find the general solution ODE pls help

**tak** · April 24th 2008, 02:47 AM

$d^2y/dx^2 - dy/dx - dy= 3e^-x + 10sinx -4x$

please check if my answer are correct

$y=c1e^-x + c2e^2x -xe^-x+ cosx +3sinx+2x$

**Isomorphism** · April 24th 2008, 03:02 AM

Originally Posted by tak

$d^2y/dx^2 - dy/dx - dy= 3e^-x + 10sinx -4x$

please check if my answer are correct

$y=c1e^-x + c2e^2x -xe^-x+ cosx +3sinx+2x$

Hello tak,
You can substitute y back in the differential equation and see if it holds.This way you can be more independent(you can even do this in exams

)

So try it

**tak** · April 24th 2008, 08:22 AM

Does it mean I have to differentiate again?

**TheEmptySet** · April 24th 2008, 08:37 AM

Originally Posted by tak

$d^2y/dx^2 - dy/dx - dy= 3e^-x + 10sinx -4x$

please check if my answer are correct

$y=c1e^-x + c2e^2x -xe^-x+ cosx +3sinx+2x$

I think we may have a problem.

$\frac{d^2y}{dx}-\frac{dy}{dx}-y=3e^{-x}+10\sin(x)-4x$

solving the homogenious equation for the particular solution we get..

$\frac{d^2y}{dx}-\frac{dy}{dx}-y=0$

$m^2-m-1=0 \iff m^2-m+\frac{1}{4}=1+\frac{1}{4} \iff (x-\frac{1}{2})^2=\frac{5}{4} \iff x=\frac{1 \pm \sqrt{5}}{2}$

so

$y_c=c_1e^{\frac{1+\sqrt{5}}{2}x}+c_2e^{\frac{1-\sqrt{5}}{2}x}$

See what you can do from here.

**tak** · April 24th 2008, 10:29 AM

$\frac{d^2y}{dx}-\frac{dy}{dx}-2y=3e^{-x}+10\sin(x)-4x<br />$
very sorry it should be 2y

**TheEmptySet** · April 24th 2008, 10:57 AM

Originally Posted by tak

$\frac{d^2y}{dx}-\frac{dy}{dx}-2y=3e^{-x}+10\sin(x)-4x<br />$
very sorry it should be 2y

then the equation is

$m^2-m-2=0 \iff (m-2)(m+1)$

so

$y_c=c_1e^{2x}+c_2e^{-x}$

Now we need to find the particular solution

since $e^{-x}$ in the complimentry solution
the particular solution will be of the form

$y_p=\underbrace{Ax^2+Bx+C}_{forThe-4x}+\underbrace{E\cos(x)+F\sin(x)}_{forThe10\sin(x )}+\underbrace{G(e^{-x})+H(xe^{-x})}_{forTheRepeted 3 e^{-x}}$

**tak** · April 24th 2008, 01:42 PM

Hi

I Know that i need to multiply by x but do I need to keep the existing term for the last part.

**tak** · April 24th 2008, 01:55 PM

how do you know that you need to keep the G(e^-x) term. I saw some website they don't add in and some does. Can I know th rule

Math Help - find the general solution ODE pls help

LinkBack

Thread Tools

Display

find the general solution ODE pls help

does it mean i have to difficentiate again

sorry it should be 2y

multiplixity one

last part

Similar Math Help Forum Discussions

Find the general solution for...

Find the general solution

Find General Solution?

find the general solution when 1 solution is given

find the general solution..

Search Tags