Find the general solution of each of the following different

Find the general solution of each of the following differential equations: y\" + 2y\' = 3 - 25 cos(2x).

Solution

First we find solution to associated homogeneous equation y\'\' 2y\' = 0

We find solutions of characteristic equation:
r² 2r = 0
r (r 2) = 0
r = 0, r = 2

y_h = c e^(2x) + c e^(0x)
y_h = c e^(2x) + c

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Find particular solution to non-homogeneous equation: y\'\' 2y\' = 3 25 cos(2x)
using method of undetermined coefficients.

Now usually, we would set y_p = A + B sin(2x) + C cos(2x). But since constant term is part of solution to homogeneous solution, we multiply A by x

y_p = Ax + B sin(2x) + C cos(2x)
y_p\' = A + 2B cos(2x) 2C sin(2x)
y_p\'\' = 4B sin(2x) 4C cos(2x)

y_p\'\' 2 y_p\' = 3 25 cos(2x)
4B sin(2x) 4C cos(2x) 2A 4B cos(2x) + 4C sin(2x) = 3 25 cos(2x)
2A + (4B + 4C) sin(2x) (4B + 4C) cos(2x) = 3 25 cos(2x)

Matching coefficients we get
2A = 3
4B + 4C = 0
4B + 4C = 25

Solving, we get: A = 3/2, B = 25/8, C = 25/8

y_p = 3/2 x + 25/8 sin(2x) + 25/8 cos(2x)

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General solution:

y = y_h + y_p

y = c e^(2x) + c 3/2 x + 25/8 sin(2x) + 25/8 cos(2x)