Question

Solve : (D2+1) y = x2sin2x ​

Answer 1

Solution:

The given differential equation is

(D² + 1) y = x² sin2x ..... (1)

To find C.F.

Auxiliary equation is m² + 1 = 0 or, m = ± i

Therefore C. F. = C₁ cosx + C₂ sinx

To find P.I.

Since the C.F. does not include x² and sin2x, we take

yₚ = (Ax² + Bx + C) (D sin2x + E cos2x)

Then, Dyₚ

= (Ax² + Bx + C) (2D cos2x - 2E sin2x) + (2Ax + B) (D sin2x + E cos2x)

and D²yₚ

= (Ax² + Bx + C) (- 4D sin2x - 4E cos2x) + (2Ax + B) (2D cos2x - 2E sin2x) + (2Ax + B) (2D cos2x - 2E sin2x) + 2A (D sin2x + E cos2x)

= - 4AD x² sin2x - 4AE x² cos2x - 4 BD x sin2x - 4BE x cos2x - 4CD sin2x - 4CE cos2x + 4AD x cos2x - 4AE x sin2x + 2BD cos2x - 2BE sin2x + 4AD x cos2x - 4AE x sin2x + 2BD cos2x - 2BE sin2x + 2AD sin2x + 2AE cos2x

= (- 4AD) x² sin2x + (- 4AE) x² cos2x + (- 4BD - 4AE - 4AE) x sin2x + (- 4BE + 4AD + 4AD) x cos2x + (- 4CD - 2BE - 2BE + 2AD) sin2x + (- 4CE + 2BD + 2BD + 2AE) cos2x

= (- 4AD) x² sin2x + (- 4AE) x² cos2x + (- 4BD - 8AE) x sin2x + (- 4BE + 8AD) x cos2x + (- 4CD - 4BE + 2AD) sin2x + (- 4CE + 4BD + 2AE) cos2x

From (1), we get

D²yₚ + yₚ = x² sin2x

or, (- 4AD) x² sin2x + (- 4AE) x² cos2x + (- 4BD - 8AE) x sin2x + (- 4BE + 8AD) x cos2x + (- 4CD - 4BE + 2AD) sin2x + (- 4CE + 4BD + 2AE) cos2x + (Ax² + Bx + C) (D sin2x + E cos2x) = x² sin2x

or, (- 4AD) x² sin2x + (- 4AE) x² cos2x + (- 4BD - 8AE) x sin2x + (- 4BE + 8AD) x cos2x + (- 4CD - 4BE + 2AD) sin2x + (- 4CE + 4BD + 2AE) cos2x + AD x² sin2x + AE x² cos2x + BD x sin2x + BE x cos2x + CD sin2x + CE cos2x = x² sin2x

or, (- 3AD) x² sin2x + (- 3AE) x² cos2x + (- 3BD - 8AE) x sin2x + (- 3BE + 8AD) x cos2x + (- 3CD - 4BE + 2AD) sin2x + (- 3CE + 4BD + 2AE) cos2x = x² sin2x

Comparing among the coefficients, we write

- 3AD = 1 or, AD = - 1/3

- 3AE = 0 or, AE = 0

- 3BD - 8AE = 0 or, BD = 0

- 3BE + 8AD = 0 or, BE = - 8/9

- 3CD - 4BE + 2AD = 0 or, CD = 26/27

- 3CE + 4BD + 2AE = 0 or, CE = 0

Thus yₚ = AD x² sin2x + AE x² cos2x + BD x sin2x + BE x cos2x + CD sin2x + CE cos2x

= - 1/3 * x² sin2x - 8/9 * x cos2x + 26/27 * sin2x

∴ the complete solution is

y = C₁ cos2x + C₂ sin2x - 1/3 * x² sin2x - 8/9 * x cos2x + 26/27 * sin2x

NOTE:

The calculation is quite big because I did not multiply the terms of yₚ in the beginning and we have terms like AD, AE, etc. When you solve this problem at home, remember to take single constants A, B, C etc.