Math 3705/H Problems

Summer 2023

  1.   (Due 7/5/23)   Solve the following: \( \ xy' - 2y = x^3 \cos x\), \( \ y(\pi) = \pi^2.\)

  2.   (Due 7/10/23)   Solve the following: \( \ y' - x^3 y^2 = 4x^3\), \( \ y(0) = 2.\)

  3.   (Due 7/11/23)   Solve the following: \( \ xy^2 \dfrac{dy}{dx} = 2x^3 - 2x^2y + y^3. \)

  4.   (Due 7/11/23)   Solve the following: \( (\ln y + 4x) dx + \dfrac{x}{y} dy = 0 \)

  5.   (Due 7/12/23)   Solve the following: \( xy'' - y' = 2x^3; \ y(1) = 0, \ y'(1) = 2 \)

  6.   (Due 7/17/23)   Solve the following: \( y'' - 4y' - 5y = 0; \ y(-1) = 3, \ y'(-1) = 9 \)

  7.   (Due 7/18/23)   Solve the following: \( y'' + 2y' + 17y = 0; \ y(0) = 1, \ y'(0) = -1 \)

  8.   (Due 7/24/23)   Solve the following: \( y'' + y' - 2y = e^x \)

  9.   (Due 7/25/23)   Using the method of variation of parameters, find a general solution to the following: \( y'' + 4y = \sec^2(2x) \)

  10.   (Due 7/25/23)   Solve the following initial value problem: \( y''' - 4y'' + y' + 6y = 0; \ \ y(0)= -1, \ \ y'(0)= 3, \ \ y''(0)= -3 \)

  11.   (Due 8/1/23)   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 8/2/23)   Determine the Laplace transform of the function \(f(t) = t e^{2t}\cos 5t\).

  13.   (Due 8/2/23)   Determine the inverse Laplace transform of the following function. $$F(s) = \frac{7s^2-41s+84}{(s-1)(s^2-4s+13)}$$

  14.   (Due 8/7/23)   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 8/7/23)   Solve the following initial value problem using the method of Laplace transforms. $$ w'' + w = u(t-2) - u(t-4); \ \ w(0)= 1, \ \ w'(0)= 0 $$