Math 315, Tests 1 and 2
How many descriptions are there of a system of ODEs and what are they? (Don't give an explanation of them)
1. Autonomous. 2. Linear. 3. Homogeneous. 4. Coupled
When is an ODE of the form: M(x,y)dx + N(x,y)dy = 0 considered homogeneous?
If BOTH M(x,y) and N(x,y) are homogeneous functions of the SAME degree.
Is this a linear 1st order ODE, why or why not? dz/dt = tsin(z)
It is not a linear 1st order ODE because tsin(z) cannot be rewritten in the form a(t)z + b(t), no matter how a(t) and b(t) are chosen. z≠sin(z)
What is the integrating factor in the method of integrating factors? Very short answer.
It's.. Math 316 190
What's the phase plane method?
Math 315 006
What's the definition of a solution to the normal form of an ODE?
Math 315 101
What's the Existence and Uniqueness Theorem for 1st Order ODEs?
Math 315 102
Math 315 313
Math 315 314
When is a first-order differential equation considered linear?
Math 316 181
Always check if the what works?
The trivial solution works.
How do equilibrium solutions set boundaries?
They set boundaries (horizontal lines) that the solution can't cross, because once it gets to that value, the dependent variable doesn't change for all time. dP/dt =0 for all time. Math 315 109
Math 316 120
We can withdraw $500 per year for 24 years. Math 316 121 and Math 316 122
Is dy/dt = g(t) separable?
Yes because we may regard the right-hand side as g(t) · 1, where we consider 1 as a (very simple) function of y.
How can a Homogeneous ODE be converted into a separable ODE?
(Second definition of homogeneous). It can be converted into a separable ODE by a change of variable. There are two possible change of variable formulas. y = ux x = vy. Those get, dy = udx + xdu, and, dx = vdy + ydv. (You take the derivative with respect to some imaginary variable and then cancel out that variable). The final solution always has a constant C.
Solve this system of ODEs: Math 315 010
(The dx/dt ODE has no 'y's in it. Therefore, the system is considered uncoupled (NOT coupled). And the system of ODEs is linear because both ODEs are linear. Because the system is uncoupled, we can solve one of the ODEs getting us x in terms of t. Then we can plug that expression for t in the other ODE. Getting us an ODE involving only y and t.) Math 315 012
What are the three kinds of equilibrium points based on f(y) around them?
1. We say an equilibrium point y0 is a sink if any solution with initial condition sufficiently close to y0 is asymptotic to y0 as t increases. 2. We say an equilibrium point y0 is a source if all solutions that start sufficiently close to y0 tend toward y0 as t decreases. This means that all solutions that start close to y0 (but not at y0) will tend away from y0 as t increases. 3. Every equilib- rium point that is neither a source nor a sink is called a node.
What are the four steps required to draw the phase line?
1.Draw the y-line. 2. Find the equilibrium points (the numbers such that f (y) = 0), and mark them on the line. 3. Find the intervals of y-values for which f (y) > 0, and draw arrows pointing up in these intervals. 4. Find the intervals of y-values for which f (y) < 0, and draw arrows pointing down in these intervals.
When is a differential equation called separable?
A differential equation is called separable if the function f (t, y) can be written as the product of two functions: one that depends on t alone and another that depends only on y. That is, a differential equation is separable if it can be written in the form: Math 316 118
What's the definition of a linear differential equation?
A differential equation is linear if the dependent variable and its derivatives appear in first-degree only (i.e., no powers different than 1, no product of dependent-variable terms, coefficients of such Terms depend possibly only on the independent variable(s)). Equations that are not linear are called nonlinear.
What's a constant-coefficient equation?
A first-order linear differential equation is a constant-coefficient equation if a(t) is a constant. In other words, the linear equation is a constant-coefficient equation if it has the form dy/dt = λy+b(t) where λ is a constant.
The pair of equations in the predator-prey model is called what?
A first-order system (only first derivatives, but more than one dependent variable) of ordinary differential equations. The system is said to be coupled because the rates of change of R and F depend on both R and F.
What must a general solution contain (for X)?
A general solution must contain the fundamental set only, and it must have the number of vectors as the number of dimensions in the ODE. Just like a solution on the x-y plane must have a general solution in only i hat and j hat.
If you integrate both sides of y' = 1, what do you get?
A one-parameter family of solutions 𝒚=𝒕+𝑪 .
What's the singular solution?
A singular solution is one that solves the ODE (like y=0 or y=1) but no choice of constant value C in the formula of the family of solutions will give that solution (y=0 or y=1).
What's an IVP?
An ODE with an IC (Initial Condition like y(1) = 2). IVP stands for Initial Value Problem.
What's the definition of a differential equation?
An equation containing the derivatives or differentials of one or more dependent variables
What's an autonomous equation?
An equation for dy/dt that does not depend on time. For example, dy/dt = y is an autonomous equation because the equation for dy/dt or f(y) (same thing) does not depend on time, it depends only on y.
The value of P(t) at t = 0 for dP/dt =kP(t0)≠0. Is called what?
An initial condition
This pair of equations is called what? Math 315 103
An initial problem. A solution to the initial value problem is a function P(t) that satisfies both equations.
What are the four kinds of intervals.
An interval, I, can be open, closed, half-open, or infinite.
If f is continuous, it can switch from positive to negative only at what points?
At points y₀ where f(y₀) = 0, that is, at equilibrium points. Hence the equilibrium points play a crucial role in sketching the phase line. (This is assuming that f(y) is autonomous.)
x1' = x2 + 2 , x2' = x1 - x2 . Describe this system.
Autonomous cause there's no explicit t-dependence. Also it's linear, not homogeneous, and it's coupled.
Why is f(x,y) = C in an exact ODE?
Because M(x,y)dx + N(x,y)dy = 0 and we've found an M and N that satisfy: δf/δx = M, and δf/δy = N. When you substitute the partial derivatives for M and N you get: (δf/δx)dx + (δf/δy)dy = 0. The dx's and dy's cancel out, and you get df = 0. That means no change in df with either a small change in x or y. In other words, f(x,y) is constant. f=C
for dw/dt = (2−w)sin(w), if we give an initial value w(0) = 0.4, what do we know from the Existence and Uniqueness Theorem.
Because w = 0 and w = 2 are equilibrium points of this equation and 0 < 0.4 < 2, we know from the Existence and Uniqueness Theorem that 0 < w(t) < 2 for all t. Moreover, because (2 − w) sin w > 0 for 0 < w < 2, the solution is always increasing. Because the velocity of the solution is small only when (2 − w) sin w is close to zero and because this happens only near equilibrium points, we know that the solution w(t) increases toward w=2 as t →∞. Similarly, if we run the clock backward, the solution w(t) decreases. It always remains above w=0 and cannot stop, since 0<w<2. Thus as t→−∞,the solution tends toward w = 0. Math 316 164
Why can't we solve the linear system? Math 315 017
Cause we can't divide a vector by another vector. Math 315 019
How can you deal with an ODE of some higher order, specifically a second order ODE?
Convert it into a system of 1st order ODEs. Math 315 013. x1 is the dependent variable. x2 is the 1st derivative of the dependent variable with respect to the independent variable.
How can we check our answers in differential equations?
Given a function, we can test to see whether it is a solution by just substituting it into the differential equation and checking to see whether the left- hand side is identical to the right-hand side. This is a very nice aspect of differential equations: We can always check our answers! So we should never be wrong. If you have a function for y(t), take the derivative of it with respect to t and make sure it is the same as the right hand side of the ODE.
How do you find the Equilibrium solution to a system of ODEs? When can this method be applied?
Given that they are autonomous, you can use this method: If x1' = g(x1, x2) and x2' = h(x1, x2), then g(x1, x2) AND h(x1, x2) must both equal 0. Note that this is not the same as finding the ordered pairs that make g(x1, x2) = h(x1, x2) If you set them equal to each other you'll get it wrong. They have to be equal to each other but also equal to 0.
What is a homogeneous differential equation?
If b(t) = 0 for all t, then the equation is said to be homogeneous or unforced. Otherwise it's nonhomogeneous or forced.
When is a system of equations coupled?
If both the dx/dt AND dy/dt ODEs contain BOTH dependent variables. In other words, the dx/dt ODE contains y AND the dy/dt ODE contains x. Then, neither equation can be solved on its own.
What is the only way, before you solve it, can an equation be an exact differential equation?
If it can be put in the form: M(x,y)dx + N(x,y)dy = 0. It may not be given in that form so you may need to try to rearrange and put it in that form.
For a linear ODE of the form dy/dt + p(t)y = g(t), when does there exist a unique solution to it?
If p(t)y & g(t) are continuous.
What are the equilibrium solutions of the logistic model? Explain.
If we sketch the graph of the quadratic function dP/dt, otherwise known as f, we see that it crosses the P-axis at exactly two points, P = 0 and P = N. In either case we have dP/dt = 0. Since the derivative of P vanishes for all t, the population remains constant if P = 0 or P = N. That is, the constant functions P(t) = 0 and P(t) = N are solutions of the differential equation. These two constant solutions make perfect sense: If the population is zero, the population remains zero indefinitely; if the population is exactly at the carrying capacity, it neither increases nor decreases. As before, we say that P = 0 and P = N are equilibrium solutions or equilibria. The constant functions P(t) = 0 and P(t) = N are called equilibrium solutions. Math 315 108
What is one way you can check if an equation is homogeneous if it's very difficult to separate it into the differential form?
If you can't put it into the M(x,y)dx + N(x,y)dy = 0 and see that both are homogeneous of the same degree, then you can simply multiply the x and the y by 't's and see that the 't's cancel. For example, Math 315 009. If the 't's in ty/tx cancel out then it is homogeneous.
Is this a linear 1st order ODE, why or why not? dy/dt = y^2
It is not a linear 1st order ODE because y^2 cannot be rewritten in the form a(t)y + b(t), no matter how a(t) and b(t) are chosen.
What does a one-parameter family of solutions mean?
It means there's the constant of integration, referred to as C. It's given as, y = (something something) + C
How do you find the solution to an exact ODE?
It must be able to be put in the form: M(x,y)dx + N(x,y)dy = 0. Furthermore, it must be 'nice enough', meaning that the function, f, the partial derivative of f with respect to x, and the partial derivative of f with respect to y must all be continuous in some rectangle. f, δf/δx, and δf/δy must be continuous in some rectangle. δf/δx = M, and δf/δy = N. That means, df = 0. Which means f(x,y) = C. You need to find that f(x,y), such that δf/δx=M(x,y), and δf/δy=N(x,y). So you'll get 'some expression' + C1 = f. But f = C2 (another constant). So ultimately you'll get 'some expression' + C1 = C2. In other words, 'some expression' = C. You may not be able to solve for y, in which case you box it as the solution.
What kind of equation does the ODE need to be in order to do integrating factors and how do you rewrite it at first? Explain the type: linear/nonlinear, etc.
It needs to be a nonhomogeneous 1st order linear differential equation. It should be rewritten as: Math 316 186
What's a homogeneous ODE? The second definition. (genous)
It's a category of ODE. If a function f(x,y) satisfies f(tx,ty)= t^(n)f(x,y), where n is a real number, then we say that f is homogen(e)ous of degree n.
What's the Superposition Principle for linear systems? Prove it. When is the Superposition Principle true?
It's a theorem: Math 315 026. Note that the linear system MUST be homogeneous.
dy/dt = y(y-1). What's the autonomousness, the order, and linearity?
It's autonomous because dy/dt has no t dependence, it's nonlinear, and it's first order.
Classify according to type (ODE or PDE), order, and linearity. If nonlinear, identify the term or terms that make the equation nonlinear. Math 315 298
Linear 1st order ODE
Classify according to type (ODE or PDE), order, and linearity. If nonlinear, identify the term or terms that make the equation nonlinear. Math 315 303
Linear 1st order ODE
Classify according to type (ODE or PDE), order, and linearity. If nonlinear, identify the term or terms that make the equation nonlinear. Math 315 301
Linear 2nd order PDE
Classify according to type (ODE or PDE), order, and linearity. If nonlinear, identify the term or terms that make the equation nonlinear. Math 315 306
Linear 2nd order PDE
Classify according to type (ODE or PDE), order, and linearity. If nonlinear, identify the term or terms that make the equation nonlinear. Math 315 300
Linear 3rd order ODE
What's the differential form?
M(x,y)dx + N(x,y)dy = 0
What's the theorem that makes it easier to check whether ODEs are exact?
M(x,y)dx + N(x,y)dy = 0. Where M, N, δM/δy, and δN/δx are continuous AND δM/δy = δN/δx. If those two conditions are met then f is exact. It works cause M must equal δf/δx, and N must equal δf/δy. Because only then, δM/δy = δf/δyδx and δN/δx = δf/δxδy. It's the mixed derivative theorem.
Suppose that you have a cylinder with water in it, and the cylinder is draining through a small circular hole at the bottom. Solve for the height of the water in the cylinder given time since the cylinder started draining.
Math 315 001
Suppose that you have a cylinder with water in it, and the cylinder is draining through a small circular hole at the bottom. For h = 10m at t = 0. A = 100m^2 and A(naught) = 10^-4 m^2, when will the tank be empty?
Math 315 001. C = √10, and it's empty when h = 0. Set the first term in the parenthesis equal to 10. You get: t = 1.4*10^6 seconds
What are the 3 main assumptions behind Newton's Law of Cooling?
Math 315 002
Solve Newton's Law of Cooling given the assumptions.
Math 315 002, Math 315 003
Math 315 004. These are the assumptions about immediately adding cream. Solve. Math 315 007. But first try to remember the assumptions.
Math 315 005
When is a system of equations linear?
Math 315 011. If g1 AND g2 are linear functions in x AND y (both dependent variables), only then do we call the system linear. Otherwise it's non-linear.
When is a system of equations autonomous?
Math 315 011. If g1 AND g2 don't have explicit t-dependence then we call the system autonomous.
How can every linear system of ODEs be written as?
Math 315 015. But the system of equations MUST be linear. The matrix is written as A usually.
y'' - y' + y = 0. Go as far as you can in solving this.
Math 315 016, Math 315 017
How do you multiply A and (X vector)? Invent one yourself.
Math 315 018
What's the definition of a solution vector of a linear system?
Math 315 020
What's an IVP for a for linear systems?
Math 315 024
What's the Existence and Uniqueness theorem for LINEAR systems? LIke the systems we would have with a matrix
Math 315 025
What's the definition of a fundamental set?
Math 315 027
What is the definition of linear Independence/Dependence?
Math 315 028
What's the method that always works to test for linear independence?
Math 315 029, Math 315 030. We'll only go up to 3 vectors (where n = 3)
What's the general solution to X' = AX, where A is a 2x2 matrix and X is a vector?
Math 315 031. But X1 and X2 must be linearly independent of each other (form a fundamental set)
x1' = x2 + 2 , x2' = x1 - x2 . Find the equilibrium points. And graph them
Math 315 033
x1' = x2 , x2' = x1 - x2 . Find the equilibrium point.
Math 315 034
x1' = sin(x1) - x2 , x2' = x2 . Find the equilibrium point.
Math 315 035
Math 315 036, Math 315 037
Math 315 038, Math 315 039
Math 315 040
Math 315 041
Math 315 042
Math 315 043
Math 315 103. What's it's solution and what's that called?
Math 315 104. That's called the solution as well as a particular solution of the differential equation. The collection of functions P(t) = c e^(kt) is called the general solution of the differential equation because we can use it to find the particular solution corresponding to any initial-value problem.
What's the logistic population model? Is it linear, why or why not?
Math 315 105, Math 315 107
What is the graph of f(P) look like for the logistic differential equation? What does the graph indicate?
Math 315 106. It indicates that f(P) is nonlinear because its right-hand side is not a linear function of P as it was in the exponential growth model.
What are the 4 assumptions of the predator-prey model?
Math 315 110
Math 315 212, Properly show whether or not it is exact
Math 315 213, Math 315 214
Math 315 215, Solve it by the exact method
Math 315 216, Math 315 217
Math 315 218, Math 315 219
Math 315 220
Math 315 221
Math 315 222
Math 315 223
Math 315 224
Math 315 225
Math 315 226
Math 315 227, Is the system linear or nonlinear, justify. Is the system autonomous or non-autonomous, explain. Is the system coupled or uncoupled, explain. What are the equilibrium points of the system?
Math 315 228
Math 315 229
Math 315 230
Math 315 231, Math 315 232, and indicate the order of the equation
Math 315 234, 2nd
Math 315 231, Math 315 233, and indicate the order of the equation
Math 315 235, 3rd
Math 315 236, Math 315 237
Math 315 239
Math 315 236, Math 315 238
Math 315 240
Math 315 241
Math 315 242
Math 315 243, Math 315 244
Math 315 245
Math 315 246
Math 315 247
Math 315 248, Math 315 249, Math 315 250, Math 315 251
Math 315 252, Math 315 253
Math 315 254, Math 315 255, just do part 2
Math 315 256
For this ODE, determine whether it is linear or nonlinear. If it is nonlinear circle the term or terms that make it nonlinear. Also, indicate what order the ODE is. Math 315 290
Math 315 292. Second order.
For this ODE, determine whether it is linear or nonlinear. If it is nonlinear circle the term or terms that make it nonlinear. Also, indicate what order the ODE is. Math 315 291
Math 315 293. Fourth order.
Math 315 294
Math 315 295
Math 315 296
Math 315 297
Classify according to type (ODE or PDE), order, and linearity. If nonlinear, identify the term or terms that make the equation nonlinear. Math 315 299
Math 315 307
Classify according to type (ODE or PDE), order, and linearity. If nonlinear, identify the term or terms that make the equation nonlinear. Math 315 302
Math 315 308
Classify according to type (ODE or PDE), order, and linearity. If nonlinear, identify the term or terms that make the equation nonlinear. Math 315 304
Math 315 309
Classify according to type (ODE or PDE), order, and linearity. If nonlinear, identify the term or terms that make the equation nonlinear. Math 315 305
Math 315 310
Math 315 315
Math 315 316
Math 315 317. Also, find the eigenvectors and what is the general solution?
Math 315 318
What are the 4 parameters of the predator-prey model?
Math 316 111
What is the predator-prey model?
Math 316 112, Check out Math 316 111, Math 316 110. For the parameters and the assumptions.
What's the standard form for a first order differential equation?
Math 316 113
What's the usual form of an initial value problem?
Math 316 114
Find the solution to dy/dt = t/y^2
Math 316 119
Draw the phase line for dy/dt = y(1-y).
Math 316 161
Draw the phase line for dy/dt = sin(y)
Math 316 162
Draw the phase line for dy/dt = ycos(y)
Math 316 163
Graph this phase line, Math 316 165
Math 316 166
Draw the phase line and sketch of solutions for dy/dt = 1/1-y
Math 316 167. All solutions tend toward y = 1 as t increases. Because the value of dy/dt is large if y is close to 1, solutions speed up as they get close to y = 1, and solutions reach y = 1 in a finite amount of time. Once a solution reaches y = 1, it cannot be continued because it has left the domain of definition of the differential equation. It has fallen into a hole in the phase line.
Draw the phase line and sketch of solutions for this graph of f(y). Math 316 168
Math 316 169
Draw the phase line and graph of solutions for a sink.
Math 316 170
Draw the phase line and graph of solutions for a source.
Math 316 171
Draw the phase line and graph of solutions for a node.
Math 316 172
Draw f(y) versus y for a sink at y = y0
Math 316 173
Draw f(y) versus y for a source at y = y0
Math 316 174
What's the metaphor of the rope?
Math 316 175. The phase line IS THE ROPE!! We can draw the phase line (or "rope") by placing dots at the equilibrium points y = 0 and y = 1. For 0 < y < 1, we put arrows pointing up because f (y) > 0 means you climb up; and for y < 0 or y > 1, we put arrows pointing down because f(y) < 0 means you climb down.
What's the linearization theorem?
Math 316 176. f'(y₀) means df/dy. Math 316 177. We cannot make any conclusion about the classification of y0 if f ′(y0) = 0, because all three possibilities can occur, Math 316 178.
Use the linearization theorem to find the phase line around y = 0 for this function. Math 316 179
Math 316 180... y = 0 as t increases. Of course, there is the dangerous loophole clause "sufficiently close." Initial conditions might have to be very, very close to y = 0 for the above to apply. Again we did a little work and got a little information. To get more information, we would need to study the function h(y) more carefully.
What's the linearity principle for homogeneous equations?
Math 316 184, Math 316 185
What do you do after you notice the left hand side looks like the product rule (integrating factors). Get the new differential equation.
Math 316 187
Once you have the new differential equation get the last step (integrating factors).
Math 316 188, note that we can't just cross out the u(t). Now we just need to find the integrating factor.
Find the general solution to this equation. Math 316 191
Math 316 192
Is dy/dt = ty^2 + 5 autonomous? Why or why not?
No it's not autonomous because there's t dependence.
Do you ignore the constant when you multiply both sides of the original equation by u(t) and integrate?
No, there it's important to include the constant of integration on the right-hand side. If you omit that constant, you will be computing just one solution to the nonhomogeneous equation (The solution where the constant is 0), rather than the general solution.
Is the fundamental set of a general solution unique?
Not necessarily. Someone can come up with another general solution for the same ODE.
What kind of functions are homogeneous (genous) functions, generally?
Often, homogeneous functions are polynomials.
What does a first-order equation contain?
Only first derivatives of the dependent variable
On what interval of time can a solution to an IVP exist?
Only on an interval where y is defined and doesn't go to infinity in finite time, like the tangent function does. So automatically if you see 1/t or 1/(t-2) that tells you that the interval cannot extend to those points. The interval of time also must include the initial condition.
What are parameters?
Parameters are quantities that do not change with time (or with the independent variable) but that can be adjusted (by natural causes or by a scientist running the experiment). For example, if we are studying the motion of a rocket, the initial mass of the rocket is a parameter.
What's a homogeneous linear system?
Same as Math 315 015, but F = 0 vector. It's very similar to the original definition of homogeneous. Note the e in homogeneous.
What's the equilibrium solution to dP/dt = kP, and why is it the equilibrium solution?
Since dP/dt = kP for some constant k, d P/dt = 0 if P = 0. Thus the constant function P(t) = 0 is a solution of the differential equation. This special type of solution is called an equilibrium solution because it is constant forever. In terms of the population model, it corresponds to a species that is nonexistent.
What's an equilibrium solution?
Solutions for y that make dy/dt =0 for all time are called equilibrium solutions for some function, dy/dt =f(t, y)
What does the integrating factor must satisfy? Derive to the end.
Technically there's a constant in front of the e but we only need one solution so we can safely get rid of the constant, also the absolute value signs can go away for the same reason. Math 316 189
If we get the particular solution to the IVP, then what happens to the one-parameter family of solutions to the ODE?
The IC reduces the one-parameter family of solutions down to the actual value for C and we get what's called the particular solution as opposed to the general solution we had before.
Which is homogeneous and which is nonhomogeneous? Math 316 183
The bottom one is nonhomogeneous, while the top one is homogeneous.
Check that this vector is a solution to this system. And describe the system. Math 315 023
The column vector has differentiable xi that satisfy the linear system, therefore it is a solution vector to the linear system. The system of ODEs is linear, coupled, autonomous, and homogeneous (homogeneous because there are no terms that don't involve the dependent variables x1 and x2).
Check that this column vector is a solution vector to this linear system? Also describe the linear system. Math 315 021
The column vector has differentiable xi that satisfy the linear system. The linear system is linear, coupled, autonomous, and homogeneous (homogeneous because there are no terms that don't involve the dependent variables x1 and x2). Math 315 022
What's another word for the general solution and what does it refer to?
The complete solution. It refers to all possible solutions.
What are the constant functions for dy/dt = y(1-y)
The constant functions are when f(y) = 0 and that happens precisely when y = 0 and y = 1. y₁(t) = 0 and y₂(t) = 1 are the equilibrium solutions for the equation.
In order for the ODE to be nonlinear, the power has to be on the what not the what?
The dependent variable, not the independent variable or something else. So on y not x or t.
Describe this system of ODEs: Math 315 010. Don't solve, just describe.
The dx/dt ODE has no 'y's in it. Therefore, the system is considered uncoupled (NOT coupled). And the system of ODEs is linear because both ODEs are linear. Because the system is uncoupled, we can solve one of the ODEs getting us x in terms of t. Then we can plug that expression for t in the other ODE. Getting us an ODE involving only y and t.
What can the graphs of solutions cannot cross?
The graphs of the equilibrium solutions.
What's the definition of the order of a differential equation?
The highest order derivative appearing in the equation.
The equilibrium points are extremely important in understanding the what of solutions?
The long-term behavior of solutions.
What's the difference between the particular solution and the singular solution?
The particular solution gives a value to the C, while a singular solution is a solution to the ODE that you can't find using any form involving C.
The general solution is in contrast to the what?
The particular solution, where certain parameter values are specified. Like C is defined as '2' instead of left as just any constant.
What is special about the slopes in the slope field of an autonomous differential equation?
The slopes are parallel along horizontal lines, (because dy/dt is the same for all t and only depends on y). Therefore, if we know the slope field along a single vertical line t = t₀, then we know the slope field in the entire t y-plane. This one line is called the phase line for the autonomous equation.
What's the general solution to (y''')^6 + 3y^4 = 0?
The trivial solution is the only solution (y=0). Because there are only even powers which means there are two positive (or 0) numbers added to each other.
Whenever some variable is included in a denominator then what?
Then the variable(s) cannot be such that the denominator equals 0. You MUST write D(x,y) ≠ 0. Where D(x,y) is the actual expression of the denominator. Don't actually write D(x,y) just write the expression in the denominator ≠ 0. so x^2 + 2y≠0 if x^2 + 2y was, at some point, in the denominator.
What's the general solution to (y')^2 + 1 = 0?
There is none, it has no real solutions.
What are equilibrium points?
They are points on the y axis where there's an equilibrium solution. So for dy/dt = y(1-y), the equilibrium points are at 0 and 1 on the y axis.
In this general solution, what is the case with X1 and X2? What is that called? Math 315 032
They form a fundamental set, that is, they are linearly independent of each other.
What kind of problem is this and what's the general solution and particular solution? Math 316 115
This is the general solution or the one parameter family of solutions: Math 316 116. And this is the particular solution, Math 316 117.
What integration methods do you need to know?
U-sub and integration by parts, as well as the basic integration rules like for ln(x) and e^x.
What are we looking for in an initial value problem?
We are looking for a function y(t) that is a solution of the differential equation and assumes the value y0 at time t0. Often, the particular time in question is t = 0 (hence the name initial condition), but any other time could be specified.
If a function f(t,y) is well behaved enough in a IVP then what?
We get a one-parameter family of solutions to the ODE. The IC "nails down" a numerical value to the constant of integration generated in solving the ODE... i.e., it specifies a particular member of the one-parameter family of solutions generated by the ODE. This solution is called a particular solution to the IVP. We use the Existence and Uniqueness Theorem to figure out if f(t,y) is well behaved enough.
What's a system of ODEs
When you have more than one dependent variable. For example, both x(t) and y(t) could be dependent variables (with t as the independent variable) and you would have a 'system of ODEs' just like we have a 'system of equations' in algebra. Two ODEs for two dependent variables, three ODEs for three dependent variables, etc.
Is this a linear 1st order ODE, why or why not? Math 316 182
Yes because it's of the form, dy/dt = a(t)y + b(t) where a(t) and b(t) are arbitrary functions of t. a(t) and b(t) can also be constant functions of t, for example, a(t) = 2 and b(t) = 8 making dy/dt = 2y+8. And a(t) and b(t) can both be 0.
Is dy/dt = h(y) separable?
Yes because we may regard the right-hand side as h(y) · 1, where we consider 1 as a (very simple) function of y. It is also said to be autonomous because dy/dt is not a function of t (note the h(y) not h(y,t) or h(t)). dy/dt does not depend on time.
Do you ignore the constant when you calculate the integrating constant why or why not?
Yes you ignore the constant because you only need one integrating factor.
Can a differential equation have solutions that look very different from each other algebraically?
Yes. But that doesn't mean every function is a solution. Given a function, we can test to see whether it is a solution by just substituting it into the differential equation and checking to see whether the left- hand side is identical to the right-hand side. This is a very nice aspect of differential equations: We can always check our answers. So we should never be wrong.
What lines can you draw on a graph of a system of equations when you're finding the equilibrium point?
You can draw the lines (straight lines for a linear system of ODEs) that are called the 'x1 nullcline', 'x2 nullcline', etc. For an xn nullcline, the line maps out the points where the xn' (the derivative of xn with respect to the independent variable, call it t) equals 0. In that case, you can think of the velocity in the xn direction is always 0. And is either in an/the other direction or 0 in all/both directions.
Go as far as you can in solving this: d^2y/dt^2 = -g(t)R^2/y^2. Also describe.
You can't put it in the phase 1 form because it's not a linear system. Math 315 014. The system is nonlinear and coupled. It's considered phase 2.
What could go wrong if you just get the general solution to a differential equation?
You could be missing the equilibrium solution: The solution to y where dy/dt =0 forever (for all time). The equilibrium solution doesn't always appear in the general solution so you should aways check it just in case. Check it by setting dy/dt =0 and solve for y. Or you could be missing the trivial solution.
How many solutions are there to this equation? y' = 1
You integrate both sides and get y = t+C. There are an infinite possible C so there are an infinite number of solutions. This is called the one-parameter family of solutions.
What do you need to do to find an inflection point for a function between two equilibria.
You need to find the second derivative of y with respect to t. You need to recognize that y' is either positive or negative the whole time. Instead of plugging in the function f(y) for y', just put y' and think of it as a positive number. Now, all you need to do is find the value of y that switches
What equations do you use in a tank exercise that has a variable area?
You use dV/dh = A(h), dV/dt = (dV/dh) (dh/dt), and dV/dt = A(h) (dh/dt)
In dy/dt = a(t)y + b(t) what special thing can happen with a(t) and b(t)?
a(t) and b(t) can both be 0. And it's still a linear 1st order differential equation.
What's the integral of 1/(1+y^2) dy
arctan(y) + C
What makes an Ordinary differential equation, 'ordinary'?
because it does not contain partial derivatives
For the logistic differential equation, if the initial population is less than 0 (P(0) < 0), then what's f(P) going to be?
d P/dt = f (P) < 0. Again we see that P(t) decreases, but this time it does not level off at any particular value since d P /d t becomes more and more negative as P(t) decreases.
For the logistic differential equation, if the initial population is greater than N (P(0) > N), then what's f(P) going to be?
dP/dt = f(P) <0. And the population is decreasing. When the population approaches the carrying capacity N, dP/dt approaches zero, and we expect the population to level off at N .
If the population of a town is 10,000 people and is increasing by 3% every year, what's the ODE of the situation and what's the solution of the ODE?
dP/dt = kP where k=0.03 and t is measured in years. The solution of the ODE is P=Ce^kt, where C is the population of the town at the time 0 years (when t=0).
What do you notice when you have the equation written the correct way before integrating factors?
dy/dt + g(t)y= b(t). We notice that the form of the left hand side looks somewhat like what we get when we differentiate using the product rule.
Show using symbols what a 1st Order IVP would look like.
dy/dt = f(t,y) Thats the ODE. y(t₀) = y₀ That's the IC. Written together, they make an IVP
Autonomous equations are differential equations of what form?
dy/dt = f(y)
What's the form of an autonomous differential equation?
dy/dt = h(y). (There is no t dependence in dy/dt)
What's the normal form for a first order ODE?
dy/dx = f(x, y) You can interchange x with t. That just indicates the independent variable.
For the logistic differential equation, if the initial population lies between 0 and N, then what's f(P) going to be?
f(P) > 0. In this case the rate of growth d P /d t = f ( P ) is positive, and consequently the pop- ulation P(t) is increasing. As long as P(t) lies between 0 and N, the population con- tinues to increase. However, as the population approaches the carrying capacity N, d P /d t = f ( P ) approaches zero, so we expect that the population might level off as it approaches N.
In the existence and uniqueness theorem, what has to be continuous for what?
f(t, y) and δf/δy have to be continuous for all y and all t.
Describe this situation. f(y) = y(y-1)(y-3), y(0) = 2
f(y) and δf/δy are continuous for all t and y, the solution is trapped between 1 and 3. But there are no boundaries so t can be (-∞,∞) for y (1, 3). f(y) is positive in this range so y is approaching 1 as t increases.
In the logistic differential equation, when does P(t) increase?
if 0 < P < N
In the logistic differential equation, when does P(t) decrease?
if P > N or P < 0
The intervals on the phase line with upward-pointing arrows correspond to what, and those with downward-pointing arrows correspond to what?
increasing solutions, decreasing solutions.
The derivative f ′(y0) tells us the behavior of the best what?
the best linear approximation to f near y₀. If we replace f with its best linear approximation, then the differential equation we obtain is very close to the original differential equation for y near y0.
For dy/dt = f(y), where f (y) is continuously differentiable for all y, if f (y(0)) < 0, then what?
then y(t) is decreasing for all t and either y(t) → −∞ as t increases or y(t) tends to the first equilibrium point smaller than y(0). f(y) is autonomous
For dy/dt = f(y), where f (y) is continuously differentiable for all y, if f (y(0)) > 0, then what?
then y(t) is increasing for all t and either y(t) → ∞ as t increases or y(t) tends to the first equilibrium point larger than y(0). f(y) is autonomous
To draw the phase line for the differential equation dy/dt = f (y), we need to know two basic things, what are they?
we need to know the location of the equilibrium points and the intervals over which the solutions are increasing or decreasing. That is, we need to know the points where f (y) = 0, the intervals where f (y) > 0, and the intervals where f (y) < 0.
A solution of the differential equation is a function of the independent variable that, when substituted...
when substituted into the equation as the dependent variable, satisfies the equation for all values of the independent variable. That is, a function y(t) is a solution if it satisfies dy/dt = y′(t) = f (t, y(t)). This terminology doesn't tell us how to find solutions, but it does tell us how to check whether a candidate function is or is not a solution. For example for the simple differential equation: dy/dt = y. We can check that the function y(t) = 3e^t is a solution. Whereas y(t) = sin(t) is not a solution.
What's the general solution to y'' + y = 0
y = C1cos(x) + C2sin(x)
What's the general solution to y'' - y = 0?
y = C1e^x + C2e^-x
For dy/dt = f(y), where f (y) is continuously differentiable for all y, if f (y(0)) = 0, then what?
y(0) is an equilibrium point and y(t) = y(0) for all t. Cause dy/dt is a function of y only, and does not depend on t. f(y) is autonomous
What's the normal/canonical form of an ODE?
y^n = f(x, y, y',... , y^(n-1) Basically it's the ODE where the largest derivative of the dependent variable is isolated. Where n is the order/highest derivative - (it's an nth order ODE) - of the ODE. We've solved for the dependent variable to the highest power.
