Question

Solve the differential equation (D⁴ + 16D²)y = x² + 5

Answer 1

Given (D⁴ + 16D²)y = x² + 5. …(1)

A.E., D⁴ + 16D² = 0 i.e. D² (D² + 16) = 0

or D = 0, 0, ±4i … (2)

Thus, the complementary function,

C.F. = (c₁ + c₂x) + (c₃cos 4x + c₄ sin 4x) … (3)

For particular integral,

Hence complete solution becomes,

y = (c₁ + c₂x) + (c₃ cos 4x + c₄sin4x) + x²/64[x²/3 + 39/4]