# 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..

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

Form

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

and I need to find the complete general equation corresponding to

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

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..

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

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

Quote:

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

Form

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

and I need to find the complete general equation corresponding to

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

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.