# Thread: Some more Differential Equations

1. ## Some more Differential Equations

The question states to find the solution of the following -

$\displaystyle \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 10y = 20 - e^2x$

I found the root, it was 3+- i

Im not too sure what to do after this, any help would be appreciated.

2. Hello, brd_7!

Find the solution: .$\displaystyle \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 10y \;= \;20 - e^{2x}$

I found the root, it was $\displaystyle 3 \pm i$

I'm not too sure what to do after this.

Those roots give us the homogeneous solution: .$\displaystyle y \;=\;e^{3x}\left(C_1\cos x + C_2\sin x\right)$

Now we must find the particular solution . . .

I conjectured that the solution is of the form: .$\displaystyle y_p \:=\:A + Be^{2x}$

. . and found it to be: .$\displaystyle y_p \;=\;2 - \frac{1}{2}e^{2x}$

The solution is: .$\displaystyle y \;=\;e^{3x}\left(C_1\cos x + C_2\sin x\right) + 2 - \frac{1}{2}e^{2x}$

3. Originally Posted by brd_7
[snip]
$\displaystyle \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 10y = 20 - e^2x$

I found the root, it was 3+- i

Im not too sure what to do after this, any help would be appreciated.
[snip]
Then you need to read this.

4. Ok, i managed to prove it all myself.. I just didnt know what to do because of the extra '20' term on the RHS.. now i know that you just put an A to represent the constant makes it a whole lot easier. Thank you!