Math 4430 – Practice Problems – Linear Equations of Second
Order.

1. Find the general solutions of the following equations:

(1) y'' + 3y' + 2y = 4

(2) y'' + 3y' + 2y = 12e^{t}

(3) y'' + 3y' + 2y = sin t

(4) y'' + 3y' + 2y = cos t

(5) y'' + 3y' + 2y = 1 + e^{-t} + sin t

(6) y'' − 2y' + 2y = e^{t }sin t

(7) y'' − 2y' + y = 6e^{t}

(8) y'' + y' + y = t^{2}

(9) y'' + 4y = t sin 2t

(10) y'' + y' = t + sin 2t

(11) 2y'' + 3y' + y = e^{-3t}

(12) y'' + y' = t + sin 2t

2. Solve the following initial value problems using Laplace transform:

(1) y'' + 3y' + 2y = 4; y(0) = 3, y'(0) = 1.

(2) y'' + 3y' + 2y = 4; y(0) = 4, y'(0) = −1.

(3) y'' + 3y' + 2y = e^{t}; y(0) = 1, y'(0) = −1.

(4) y'' − 5y' + 6y = e^{t}; y(0) = 1, y'(0) = 1.

(5) y'' − y = e^{2t}; y(0) = 0, y'(0) = 1.

(6) y'' − 4y = e^{t}; y(0) = 1, y'(0) = 1.