Differential Equations Solve the following differential equa

Differential Equations:

Solve the following differential equations by using the method of undetermined coefficients for finding the particular solution.

y\'\' + 4y = sin(5x)

Much appreciation.

Solution

Let d/dx = D

=> y\'\' + 4y = sin (5x) ------------------ (1)

=> D²y + 4y = sin(5x)

or (D² + 4)y = sin(5x)

Its Complementary function is given by putting D² + 4 = 0

=> D=2i,-2i

CF = c₁ cos 2x + c₂ sin 2x

For Particular integral

let trial solution be y* = A sin (5x) +B cos(5x) -------- (2)

=> D² (y*) = D (5A cos (5x) - 5B sin (5x))

= -25A sin(5x) - 25B cos(5x) ------------------ (3)

Putting (2) and (3) in (1) as y* must satisfy equation (1)

we get

-25A sin(5x) - 25B cos(5x) + 4A sin(5x) + 4B cos(5x) = sin(5x)

or -21A sin(5x) - 21B cos(5x) = sin(5x)

comparing coefficeints of sin(5x) and cos(5x) we get

-21A=1 and -21B = 0

=> A = -1/21 , B =0;

So PI = y* = -1/21 sin(5x)

Complete solution => y = c₁ cos(2x) + c₂ sin(2x) - 1/21 sin(5x).