Question

Solve by the method of undetermined coefficients; (D³ + 2D² – D –2) y = x² + e^x.

Answer 1

A.E. is 

m³ + 2m² – m – 2 = 0 

i.e., (m + 2) (m – 1)(m + 1) = 0 

So that m = ± 1, – 2

Hence C.F. is C.F. = C₁ e^x + C₂e^–x + C₃ e^–2x 

Note that e^x is common in C.F. and the R.H.S. of the given equation. 

Therefore P.I. is of the form y_p = a + bx + cx² + dxe^x ...(1) 

We have to find a, b, c and d such that  y′′′_p + 2y"_p - y'_p - 2y_p = x² + e^x ...(2)