Calc 4 Final Exam

Find e^At using Laplace transform

1. Compute (sI-A) and it's inverse 2. decompose entries 3. Inverse Laplace

Use the convolution theorem to evaluate ∫

1. Find F(s) & G(s) (if no F(s) it's 1) 2. Laplace of both 3. inverse Laplace of answer

For each of the following matrices find a fundamental set of real valued solutions for y' = Ay (repeated root)

1. Find eigenvalues 2. if the matrix is all zeros, any nonzero vector is an eigenvector 3. real set of solutinos is {e^lambda(t)v1, e^lambda(t)v2}

For each of the following matrices find a fundamental set of real valued solutions for y' = Ay (complex root)

1. Find eigenvalues 2. plug back into matrix 3. find K1 matrix 4. find X1 --> K1 e^lambda(t) 5. Use euler's rule to get equation e^lambda(t) = e^t(cos(ßt)+isin(ßt)) 6. multiply (5) by K1 spread out (real numbers vector + imaginary vector) 7. Use Complex root config e^t [cos(ßt)e1-sin(ßt)e2] + e^t[cos(ßt)+sin(ßt)] 8. Multiply out to get real solutions

For each of the following matrices find a fundamental set of real valued solutions for y' = Ay (distinct root)

1. Find eigenvalues 2. REF of matrix 3. y1(t) = e^lambda(t)v1 4. (A-I)u = v1 --> Find u 5. e^lambda(t)(u+tv1)

Solve the initial value problem y''+y'-6y = 12x; y(0) = 0 y'(0) = 2

1. Find homogeneous solution (Change to M's) [yh] 2. Find particular solution [yc] 3. y = yh + yc 4. notice conditions, find constants 5. plug constants back into y

Use variation of parameters to find a particular solution to y'' - 2y' + y = 14 x^(3/2)e^x

1. Find homogenous solution (change to M's) 2. Wronskian (Matrix w/ y1 y1' & y2 y2') / determinant of matrix 3. Use variation of parameters equation to get v'1 & v'2 4. integral of v'1 & v'2 to get v1 and v2 5. yp = v1y1 + v2y2

Solve y'' + y =∂(t-π)

1. S equivalent of equation 2. e^-st on right side 3. Get Y(s) alone 4. Inverse Laplace w/ unit step function

solve the Bernoulli equation explicitly for y 7x(dy/dx)-2y= x^2/y^6

1. Setup equation so nothing is on dy/dx 2. find w = y^1-(n) (n is power of y on right side) 3. solve for y 4. derivative of w, get dy/dx by itself 5. substitute w's for y's 6. simplify and find integrating factor 7. integrate and solve for y

Find AB

1. Split B into columns 2. Multiply each column by A 3. Result in matrix

Find all eigenvalues and eigenvectors

1. Subtract lambda from A 2. find values 3. plug values into matrix 4. note free variables and make vectors

Solve the integral equation y(t) = sint - 2∫y(T)cos(t - T) dT

1. Use convolution theorem 2. Get Y(s) by itself 3. take inverse of Y(s)

Find the inverse Laplace transform of F(s) = e^-2s/s-3 and express as a piecewise function

1. Use equation from formula f(t-a)U(t-a) 2. Inverse Laplace of F(s) 3. Write piecewise of f(t)

Find a formula (involving an integral) for the solution of the initial value problem y'' +4y' +4y = g(t), y(0) = 0 y'(0) = -2

1. Write out s equivalent to equation 2. Solve for Y(s) 3. Laplace each term 4. Use F(s)G(s) equation to get integral

Give a set of linearly independent real-valued solutions of the differential equation y''-6y'+13y = 0

1. change DE to M's, homogenous solution 2. Factor or use quadratic formula 3. Distinguish the type of root 4. Write equation 5. write set of solutions

Tank problem

1. dA/dt = FinCin - FoutCout 2. Find in and out *A / (num of gallons +/- increasing or decreasing) 3. initial condition of salt that has been dissolved

without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation y'' + 3y' - 4y = 5e^-4t is

1. find homogeneous solution (change to M's) 2. notice your particular solution can't be in your homogenous 3. Constant times particular and add variable so it's not in homogeneous so it's not in homogeneous yp = Ate^(-4t) [powers matter]

Period function

1. find period 2. write piecewise function of period 3. unit step of piecewise 4. Laplace of all terms 5. use periodic function equation 6. simplify

The autonomous differential equation dx/dt = (x-1)^2x has a solution that is

1. find values 2. make graph 3. find positive or negative of each condition to determine if it's increasing or decreasing

Solve the linear differential equation x(dy/dx)+2y = 4x^2 explicitly

1. get dy/dx by itself 2. find integrating factor (always on y) P(x) 3. take integral of the integrating factor 4. e^(P(x)) 5. mulitply both sides by integrating factor (4) (left side P(x)*y) 6.take integral of both sides 7. solve for y

if -1 is an eigenvalue of A, find all corresponding eignevalues

1. input eigenvalue 2. reduce to echelonform 3. # free variables is number of vectors 4. parametric vector form

Find a solution to the initial value problem if y(-1) = 3 and give the largest interval of existence for the solution

1. insert values into the equation 2. Solve for C 3. take derivative of y

One of the following equations is exact. Determine which one and find implicit solution

1. partial derivative of m & n (opposite of dx or dy) 2. integral of Mdx = M + g(y) 3. take y partial derivative of ^ to get g'(y) 4. integral of g'(y) to get g(y) 5. f(x,y) = M + g(y) = 0 (implicit equation)

unit-step function

1. pay attention to conditions (x1U(t-a1)- x1U(t-a2)) + x2U(t-a3))

Check your solution by plugging back into the original differential equation

1. plug in y' into original differential equation 2. should equal right side

Find inverse Laplace transform of F(s)

1. s on top has to be same as s on bottom 2. add k to numerator 3. separate and use formula sheet to get f(t)

solve the differential equation explicitly for y

1. separate common terms 2. take integral of both sides 3. solve for y

Find an explicit solution of the homogeneous differential equation and solve the initial value problem

1. substitution y = vx | dy = vdx + xdv 2. Make separable (similar variables) 3. integrate both sides 4. substitute v = y/x 5. solve for y 6. solve for c w/ conditions 7. put c back into y (5)

Solve the system by Gaussian or Gauss-Jordan elimination using the algorithm demonstrated in class. Write your solution in parametric vector form and give at least 2 particular solutions

1. take system to augmented matrix 2. REF 3. take free variables and make parametric vector form 4. choose two values for free variable and add them using the parametric vector form

Find the solution of the initial value problem if A is the matrix in partB and y(0) = [3 -2]

1. write general solution y(t) = C1[v1] + C2[v2] 2. plug in values from y(0) = [3 -2] 3. simplify

One solution of the differential equation x^2y''+xy'-y = 0 is y1(x) = x. Find another solution that is not a scalar multiple of y1(x), then write the general solution to the problem

1. y2(x) = y1(x) ∫e^(-∫P(x) dx) / (y1(x))^2 2. divide to get highest order alone 3. get integrating factor from y' 4. solve integral equation 5. write general solution y = c1y1 + c2y2 (y1 is given)

nonsingular

A is invertible A is row equivalent to identity matrix det(A) is NOT equal to 0

underdamped system

a < k

undamped system

a = 0

critical damped system

a = k

overdamped system

a > k

Newton's Law of Cooling/Warming

dT/dt = k(T-Tm) T(0) = condition

singular

det(A) IS equal to 0 Ax = 0 has many solutions A is NOT invertible

Conjugate Complex Roots

e^(iø) = cosø+isinø or y = C1(e^(ax)cos(ßx)+C2(e^(ax)sin(ßx)

non-linear

if y is multiplied by anything yy' = x

linear

if y is not multiplied by anything y''+y' =tanx

exponential decay

sign of k has to be negative k < 0

exponential growth

sign on k has to be positive k > 0

A model of a spring/mass system is

x''+2ax'+k^2x = 0

distinct real roots

y = C1(e^M1x)+C2(e^M2x)

repeated real roots

y = C1(e^M1x)+C2(xe^M1x)

What are all values of c for which y =c is constant solution of y' = (y-3)2sin5y

y = c, make the solution 0