Question

Solve the differential equations d²y/dx² + a²y = cosec ax

Answer 1

The given equation in its symbolic form is (D² + a² )y = cosec ax 

Thus, the auxiliary equation becomes D² + a² = 0 i.e. D = ±ia

∴ C.F. = (c₁cos ax + c₂sin ax)

For P.I = 1/D² + a² cosec ax

= 1/D² -(-a²) cos ec ax

= 1/(D + ia)(D - ia)cosec ax

= 1/2ia(1/D -ia - 1/D + ia) cosec ax(By partial Fractions)

On the same lines, changing i to –i, we get

1/D + ia cosec ax = e^-iax[log(sin ax/a) + ix]

Using (1) and (2),

Hence the complete solution is

y = (c₁ cos ax + c₂ sin ax) + 1/a (log(sin ax)/a sin ax - x cos ax)