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Linear Equations of Second Order

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 = 12et

(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 = et sin t

(7) y'' − 2y' + y = 6et

(8) y'' + y' + y = t2

(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 = et; y(0) = 1, y'(0) = −1.

(4) y'' − 5y' + 6y = et; y(0) = 1, y'(0) = 1.

(5) y'' − y = e2t; y(0) = 0, y'(0) = 1.

(6) y'' − 4y = et; y(0) = 1, y'(0) = 1.

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