# Variation of parameters question

• May 26th 2009, 08:27 AM
dankelly07
Variation of parameters question
Can someone help get me started with this I'm not sure where to begin..

$
\begin{gathered}
x\frac{{dy}}
{{dx}} + (2 + 3x)\frac{{dy}}
{{dx}} + 3y = r(x); \hfill \\
\hfill \\
Solution \hfill \\
y = 1/x \hfill \\
with \hfill \\
r(x) = 0 \hfill \\
\end{gathered}
$

Form

$
y(x) = \frac{1}
{x}v(x)
$

and I need to find the complete general equation corresponding to

$
r(x) = 3e^{ - 3x}
$

Can someone get me started with this.. do I differentiate y(x) part and substitute that into the equation?(Wondering)
• May 26th 2009, 12:03 PM
HallsofIvy
Quote:

Originally Posted by dankelly07
Can someone help get me started with this I'm not sure where to begin..

$
\begin{gathered}
x\frac{{dy}}
{{dx}} + (2 + 3x)\frac{{dy}}
{{dx}} + 3y = r(x); \hfill \\$

You mean $\frac{d^2y}{dx^2}+ (2+3x)\frac{dy}{dx}+ 3y= r(x)$

Quote:

$\hfill \\
Solution \hfill \\
y = 1/x \hfill \\
with \hfill \\
r(x) = 0 \hfill \\
\end{gathered}
$

Form

$
y(x) = \frac{1}
{x}v(x)
$

and I need to find the complete general equation corresponding to

$
r(x) = 3e^{ - 3x}
$

Can someone get me started with this.. do I differentiate y(x) part and substitute that into the equation?(Wondering)
Because this is a second order equation, you will need two independent solutions to the homogeneous differential equation. Try putting y= v(x)/x into the differential equation. That reduces to a first order, separable equation so that you can find v and then another y solution.

After you you have done that, you can use those solution in variation of parameters.