# Math 3705/H Problems

## Summer 2022

1.   (Due 7/6/22)   Solve the following: $$\ xy' - 2y = x^3 \cos x$$, $$\ y(\pi) = \pi^2.$$

2.   (Due 7/11/22)   Solve the following: $$\ y' - x^3 y^2 = 4x^3$$, $$\ y(0) = 2.$$

3.   (Due 7/11/22)   Solve the following: $$\ xy^2 \dfrac{dy}{dx} = 2x^3 - 2x^2y + y^3.$$

4.   (Due 7/12/22)   Solve the following: $$(\ln y + 4x) dx + \dfrac{x}{y} dy = 0$$

5.   (Due 7/13/22)   Solve the following: $$xy'' - y' = 2x^3; \ y(1) = 0, \ y'(1) = 2$$

6.   (Due 7/18/22)   Solve the following: $$y'' - 4y' - 5y = 0; \ y(-1) = 3, \ y'(-1) = 9$$

7.   (Due 7/18/22)   Solve the following: $$y'' + 2y' + 17y = 0; \ y(0) = 1, \ y'(0) = -1$$

8.   (Due 7/25/22)   Solve the following: $$y'' + y' - 2y = e^x$$

9.   (Due 7/26/22)   Using the method of variation of parameters, find a general solution to the following: $$y'' + 4y = \sec^2(2x)$$

10.   (Due 8/1/22)   Solve the following initial value problem: $$y''' - 4y'' + y' + 6y = 0; \ \ y(0)= -1, \ \ y'(0)= 3, \ \ y''(0)= -3$$

11.   (Due 8/3/22)   Using the definition of the Laplace transform, find the Laplace transform of the following function. $$f(t) = \begin{cases} e^{-t}, &\text{if $$0 < t < 2,$$} \\ 1, &\text{if $$2 < t.$$} \end{cases}$$

12.   (Due 8/3/22)   Determine the Laplace transform of the function $$f(t) = t e^{2t}\cos 5t$$.

13.   (Due 8/8/22)   Determine the inverse Laplace transform of the following function. $$F(s) = \frac{7s^2-41s+84}{(s-1)(s^2-4s+13)}$$

14.   (Due 8/8/22)   Solve the following initial value problem using the method of Laplace transforms. $$y'' - 4y = 4t - 8 e^{-2t}; \ \ y(0)= 0, \ \ y'(0)= 5$$

15.   (Due 8/9/22)   Solve the following initial value problem using the method of Laplace transforms. $$w'' + w = u(t-2) - u(t-4); \ \ w(0)= 1, \ \ w'(0)= 0$$