Question

Solve by the method of variation of parameters d²y/dx² + y = cosecx.

Answer 1

We have 

(D² + 1)y = cosec x 

A.E. is m² + 1 = 0 ⇒ m² = – 1 ⇒ m = ± i 

Hence the C.F. is given by 

∴ y_c = C₁cos x + C₂ sin x ...(1) 

y = A cos x + B sin x ...(2) 

be the complete solution of the given equation where A and B are to be found. 

The general solution is y = Ay₁ + By₂ 

We have y₁ = cos x and y₂ = sin x 

y′₁ = – sin x and y′₂ = cos x 

W = y₁y'₂ - y₂y'₁ 

= cos x . cos x + sin x . sin x = cos²x + sin²x = 1