# just a quick particular integral question..

• May 22nd 2009, 07:34 AM
dankelly07
just a quick particular integral question..
if i have a problem...

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

would the substitution be..

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

I know it would be for = xe^-2x..
• May 22nd 2009, 07:49 AM
TheEmptySet
Quote:

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

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

would the substitution be..

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

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

Quick question is it supposed to be

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

Or what you have typed above
• May 22nd 2009, 08:11 AM
HallsofIvy
Quote:

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

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

would the substitution be..

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