Part A-

1. Assume that the given function f(x) is the right hand side of some differential equation of the second order or higher. Come up with two versions of the left hand side of the differential equation so that

(i) No resonance occurs, that is, there is no duplication of the terms between the complimentary solution and the initial template of the particular solution

(ii) Resonance occurs, that is, there is some duplication of the terms between the complimentary solution and the initial template of the particular solution

(a) f(x) = 12

(b) f(x) = x

(c) f(x) = e^-5x

(d) f(x) = sin 3x

(e) f(x) = e^-5x cos 3x

(f) f(x) = x⁶

(g) f(x) = x⁵e^-3x

(h) f(x) = x⁴sin 5x

(i) f(x) = x³e^-3xcos 5x

2. We consider the following initial value problem:

Y''' + 6y'' + 10y' = 15e^-3t + 39 cost + 30t² - 4t + 12

y(0) = 3,   y'(0) = 8,    y''(0) = 11

(a) Use methods (characteristic equation, complementary solution, particular solution) to solve the initial value problem

Be sure to show details of how you have accomplished the following steps by hand:

i. How you solved the characteristic equation to find the complimentary solution

ii. Template you used to find particular solution

iii. How you found the values of arbitrary constants to solve the initial value problem

You may use Maxima to solve any systems of linear equations you encounter, but you have to set up such systems of linear equations by hand. You want to attach printout of Maxima worksheet to your solution.

(b) Use Laplace Transform methods to solve the same initial value problem.

Be sure to show details of how you have accomplished the following steps by hand:

i. Laplace Transform of the initial value problem

ii. Partial Fractions Decomposition in preparation for inverse Laplace Transform

iii. Inverse Laplace Transform

You may use Maxima to solve any systems of linear equations you encounter, but you have to set up such systems of linear equations by hand. Attach printout of Maxima worksheet to your solution.

Part B-

This problem asks you to construct something that we would call a fundamental solution to the differential equation. Consider the differential equation

Y''' + 8y'' + 15y' = 12e-3t - 130 sin t + 45t2 - 12t + 19

First, you would have to solve the following 4 initial value problems:

1. Find function y0 that satisfies the differential equation

Y''' + 8y'' + 15y' = 12e^-3t - 130 sin t + 45t² - 12t + 19

and the initial conditions

y(0) = 0,  y'(0) = 0,  y''(0) = 0

2. Find function y₁ that satisfies the homogeneous differential equation

Y''' + 8y'' + 15y' = 0

and the initial conditions

y(0) = 1,  y'(0) = 0,  y''(0) = 0

3. Find function y₂ that satisfies the homogeneous differential equation

Y''' + 8y'' + 15y' = 0

and the initial conditions

y(0) = 0,   y'(0) = 1,   y''(0) = 0

4. Find function y₃ that satisfies the homogeneous differential equation

Y''' + 8y'' + 15y' = 0

and the initial conditions

y(0) = 0,   y'(0) = 0,  y''(0) = 1

5. Now you have an alternative version of the general solution to the original differential equation, constructed as follows:

y = y₀ + C₁y₁ + C₂y₂ + C₃y₃

Note that it is now quite easy to find the constants so that y satisfies the initial conditions. To see that, use the general solution you have just found to solve the following initial value problems.

(a) Find function y that satisfies the differential equation

Y''' + 8y'' + 15y' = 12e^-3t - 130 sin t + 45t² - 12t + 19

and the initial conditions

y(0) = -3,   y'(0) = 4,   y''(0) = 5

(b) Find function y that satisfies the differential equation

Y''' + 8y'' + 15y' = 12e^-3t - 130 sin t + 45t² - 12t + 19

and the initial conditions

y(0) = 7,   y'(0) = -2,   y''(0) = 6