# Thread: just a quick particular integral question..

1. ## just a quick particular integral question..

if i have a problem...

$\displaystyle y'' + 2y' + 4 = x^2 e^{ - 2x}$

would the substitution be..

$\displaystyle y(x) = e^{ - 2x} v$

I know it would be for = xe^-2x..

2. Originally Posted by dankelly07
if i have a problem...

$\displaystyle y'' + 2y' + 4 = x^2 e^{ - 2x}$

would the substitution be..

$\displaystyle y(x) = e^{ - 2x} v$

I know it would be for = xe^-2x..
Quick question is it supposed to be

$\displaystyle y'' + 2y' + 4{\color{red}y} = x^2 e^{ - 2x}$

Or what you have typed above

3. Originally Posted by dankelly07
if i have a problem...

$\displaystyle y'' + 2y' + 4 = x^2 e^{ - 2x}$

would the substitution be..

$\displaystyle y(x) = e^{ - 2x} v$

I know it would be for = xe^-2x..
It's not clear what you are asking. You seem to be conflating "variation of parameters" and "undetermined coefficients".

Assuming you mean $\displaystyle y"+ 2y'+ 4y= x^2e^{-2x}$, which has $\displaystyle e^{-2x}$ and $\displaystyle xe^{-2x}$ as independent solutions to the associated homogenous equation, in order to use "variation of parameters" you would have to use $\displaystyle y(x)= u(x)e^{-x}+ v(x)x^2e^{-x}$. If you are referring to "undetermined coefficients", you would try $\displaystyle y(x)= (Ax^2+ bx+ C)x^2e^{-x}= (Ax^4+ Bx^3+ Cx^2)e^{-x}$.