Question

Using the method of undetermined coefficients solve d²y/dx² - 2dy/dx + 3y = x² + cosx.

Answer 1

We have 

(D² – 2D + 3)y = x² + cos x 

A.E. is m² – 2m + 3 = 0

From (1), we obtain

y′p = b + 2cx – dsin x + e cos x

y′′p = 2c – d cos x – e sin x

Hence (2) becomes,

(2c – d cos x – e sin x) – (2b + 4cx – 2d sin x + 2e cos x) + (3a + 3bx + 3cx2 + 3d cos x + 3e sin x) = x² + cos x

Comparing both sides, we obtain

2c – 2b + 3a = 0 ...(3) 

– 4c + 3b = 0 ...(4) 

3c = 1 ...(5) 

2d – 2e = 1 ...(6) 

2d + 2e = 0 ...(7) 

Solving these equations, we get