Differential Equations Exam 2
In variation of parameters what is U1 equal to?
-1 * the integral of (y2*g(t)) / W Where g(t) is given by the DE and W is the wronskian
What is the Wronskian?
A two by two matrix with y1 at the top left corner and y2 at the top right corner. Directly below them is their derivative. To find the wronskian we must first set up the matrix then take y1y'2 - y2y'1.
What is case 2 for a second order DE?
After factoring the characteristic equation and solving for r you get two imaginary roots of the form r = α +- µi. The solution in this case is y= e^αx [ c1 cos(µx) + c2 sin(µx) ]
What is case 3 for a second order DE?
After factoring the characteristic equation and solving for r you get two real roots equal to each other. The solution in this case is y= c1e^rx + c2 x e^rx
What is case 1 for a second order DE?
After factoring the characteristic equation and solving for r you get two roots r1 and r2 that are not equal to each other. The solution in this case is y= c1e^r1x + c2e^r2x
What is the frequency in mechanical vibrations?
F = Wot = Wo
How do you find the solution of a 2nd order DE set equal to zero when given two initial conditions?
First find its characteristic equation then factor and solve for r. Determine what case the roots fall under and plug r into that equation. Next apply initial conditions y(x) = ?, and y'(x) = ?. Be sure to find the derivative and then solve when plugging in the second initial condition. The goal is to find the values of C1 and C2.
How do we find the general solution using variation of parameters?
First find the characteristic equation then factor and solve for r. Identify the case function and plug in r. Let y1 = C1y1 and let y2 = C2y2 where the coefficients C1 and C2 are ignored in this case. Find the wronskian of y1 and y2 and ensure it doesn't equal zero. Solve for U1 and U2. Final answer will be y = U1y1 + U2y2.
How do you find the general solution of a non homo 2nd order DE?
First find the characteristic equation then factor and solve for r. This will be the first part of the solution. Then make a general assumption based of g(t) that we will call yp. Differentiate yp twice then sub back into the original non home DE equal to g(t). Solve for the coefficients in yp by setting all the coefficients with g(t) in them equal to g(t) and the ones that don't contain the function of g(t) equal to zero. Add yp to the first part of your solution and you should get y = C1y1 + C2y2 + y(t)
How do you find the set of y1y2 using theorem 3.2.5 for a 2nd order DE?
First find the characteristic equation, then factor and solve for r. Identify the case then plug in r. Apply the first initial condition for y1. Take the derivative of the function and apply the second condition for y1. Solve for c1 and c2 for y1. Do the same thing this time using the inital conditions for y2 using the same case function. Solvefor c1 and c2 for y2. Both y1 and y2 after solving for the coefficients will be your answers.
What does it mean to find the general solution of a 2nd order DE set equal to zero?
First find the characteristic equation, then factor and solve for r. Identify what case the roots fall under case 1, 2 or 3. Plug r into the general solution based off the case.
How do you find the fundamental set of a 2nd order DE, when given a single solution? *reduction of order*
First prove the solution is indeed a solution if need be. Let y= vy1 where v is a variable of a function and y1 is the solution given in the problem. Differentiate twice and then substitute into the DE. If done right all the V terms should cancel and you should get an equation that looks like this: a(t)V" + b(t)V' = 0 Let z = v' ; and z' = v" and solve the equation like you would a seperable or linear equation. After integrating you should get a function that has z = c1(function) Substitute v' back into the equation and integrate again getting the equation v = c1(function) + c2 Multiply this entire function by the original solution given in the problem to get your final answer.
How do you prove that two solutions form a fundamental set of solutions?
First you must prove that they are indeed solutions. Take the derivative twice and then plug into the DE. Do the same for the second solution. If they both equal zero then they are indeed solutions. Next find the wronskian of the two solutions if the wronskian DOES NOT equal zero then the two form a fundamental set.
What does it mean to have existence and uniqueness in a 2nd order DE?
If we have the equation y" + P(t)y' + Q(t)y = G(t); and the points y(t0) = yo; y'(to) = y'0 and the functions contain the value of t0, then the solution is unique and along the interval a solution is certain to exist.
What is the principle of superposition?
If y1 and y2 are both solutions to a linear homogeneous solution then so is y= c1y1 + c2y2
If a problem asks for the particular solution in a non home DE what is it asking for?
It wants you to make the assumption of g(t) and this is your yp. Solve this how you would normally; you won't be finding the characteristic equation in this sense.
What is theorem 3.2.5?
Let y1 and y2 be solutions that satisfy y1(to) = 1; y'1(to) = 0 y2(to) = 0; y'2(to) = 1 where t0 is a value given in the problem
What is a safe assumption if sin(ax) or cos(ax) = g(t) in a non home DE?
Let yp = Acos(ax) + Bsin(ax) if either functions are shown where A and B are the variable of the coefficient you must solve for and the little 'a' in front of the x is given by the function g(t)
What is a safe assumption if e^t = g(t) in a non homo DE?
Let yp = Ae^mt where A is the variable of the coefficient in front of e you must solve for and m is the cefficient in front of the exponent given by g(t)
What is a safe assumption if x + C or x^2 + x + C = g(t) in a non homo DE?
Let yp = Ax + B or Ax^2 + Bx + C where A, B, and C is the variable of the coefficient you must solve for and the number of coefficients is given by the power of x in g(t)
What is the amplitude equal to in mechanical vibrations?
R = sqrt( C1^2 + C2^2 )
What is the period in mechanical vibrations?
T = 2pi / W0
What is the phase shift in mechanical vibrations?
Tan^-1 ( sin / cos ) or Tan^-1 ( C2 / C1 )
How do you solve a 2nd order DE for existence and uniqueness?
We first start by getting y" by itself if it isn't. Identify P(t), Q(t), and G(t). Determine where each of these functions is not continuous. Draw a number line and put a little x where the function is not continuous. If t0 given by the initial conditions exist within an interval on the number line then a uniqe solution exists. What is the interval where t0 exists?
What is the case of duplication in a non homo DE?
When putting the function into its characteristic equation and then solving for r you get a root that is similar to the functions root in g(t) Ex: r= -1 and g(t) = e^-t When this happens we must multipy our duplicate assumption by t and then solve for the particular solution
What is a homogeneous 2nd order differential equation with constant coefficients?
Written in the form ay" + by' + cy = g(x) or ay" + by' + cy = 0
What is the general solution of a non homogeneous DE?
Y(t) = c1y1 + c2y2 + y(t) where the two left solutions are given by the case function and y(t) is given by g(t)
What is a characteristic equation?
ar^2 + br + c = 0 a different way of rewriting the equation of a 2nd order DE
In mechanical vibrations what is the positive direction?
downward direction
What is our standard equation for mechanical vibrations?
mU" + ΓU' + kU = f(t); u(t) Where u(t) is the displacement from equilibruim position(when the weight stops stretching the spring) and these are your intial conditions for the DE. Where m is the mass of the object Γ = forces caused by damping k = the stretch of the spring F(t) = any other external forces acting on the body *usually if the equation doesn't mention damping or f(t) then we can assume they are equal to zero*
How do you find the value of K in mechanical vibrations?
mg = kL where m is the mass given and g is the acceleration due to gravity L is the length stretched by the weight
How do you find the roots of a function if the value under the radical of the quadratic formula equals zero?
r = -b/2a , -b/2a
In variation of parameters what is U2 equal to?
the integral of (y1*g(t)) / W Where g(t) is given by the DE and W is the wronskian
When solving for mechanical vibrations what case function are you most likely to get?
u(t) = C1cos(Wo t) + C2sin(Wo t)
What does a non homogeneous 2nd order DE look like?
y" + p(t)y' + q(t)y = g(t) where g(t) does not equal zero in this case
