Question

A nonhomogeneous equation and a particular solution are given. Find a general solution for the equation. y prime prime plus 9 y prime plus 18 y equals

Answer 1

Answer:

$y_p(x) =c_1e^{-6x} + c_2e^{-3x}+ e^x + 4x^2$

Step-by-step explanation:

Given

$y" + 9y' + 18y = 24x^2 + 40x + 8 + 12e^x$ ---- (1)

$y_p(x) = e^x + 4x^2$

Required

The general solution of $y(x)$

Let

$y = e^{nx}$ be the trial solution of (1)

So:

$y" + 9y' + 18y = 0$ becomes

$n^2 + 9n + 18 = 0$

Expand

$n^2 + 6n+3n + 18 = 0$

Factorize

$n(n + 6)+3(n + 6) = 0$

Factor out n + 6

$(n + 6)(n + 3) = 0$

Split

$n +6 = 0\ or\ n + 3 = 0$

Solve for n

$n =-6\ or\ n = -3$

So:

$y = e^{nx}$ becomes:

$y = c_1e^{-6x} + c_2e^{-3x}$

$y_p(x) = e^x + 4x^2$ becomes

$y_p(x) =c_1e^{-6x} + c_2e^{-3x}+ e^x + 4x^2$

Where: $c_1$ and $c_2$ are arbitary constants