# Linear Differential Equations

• Apr 26th 2009, 01:30 PM
Nyoxis
Linear Differential Equations
Hello, I'd appreciate some help finding the general solution of the differential equation:

\frac{d^2 y}{dx^2} + 4y = \sin{x}

a^2 + 4a =0
(a + 2)^2 = 4
a = 0, a = 4

y = Ce^4x + B

But how do I find the particular integral?

Edit: Sorry it's been a while since I've Latex'd, and I'm kinda in a hurry, so I'll leave it as it is.
• Apr 26th 2009, 01:43 PM
running-gag
Quote:

Originally Posted by Nyoxis
Hello, I'd appreciate some help finding the general solution of the differential equation:

\frac{d^2 y}{dx^2} + 4y = \sin{x}

a^2 + 4a =0
(a + 2)^2 = 4
a = 0, a = 4

y = Ce^4x + B

But how do I find the particular integral?

Edit: Sorry it's been a while since I've Latex'd, and I'm kinda in a hurry, so I'll leave it as it is.

Hi

First the characteristic equation is $\lambda^2 + 4 = 0$ which leads to $\lambda = -2i$

$y = A \cos 2x + B \sin 2x$

The particular solution can be found as $C \sin x$

$-C \sin x + 4C \sin x = \sin x$ leads to $C = \frac13$