# Math Help - Non Linear homogenous differential equation

1. ## Non Linear homogenous differential equation

Just need a bit of help finishing this one, not sure if I'm going about this correctly...

$
\begin{gathered}
\frac{{dy}}
{{dx}} = \frac{{y^2 + 3xy + x^2 }}
{{x^2 }} \hfill \\
\frac{{dy}}
{{dx}} = \frac{{v^2 + 3v}}
{x} - \frac{v}
{x} \hfill \\
\frac{{dy}}
{{dx}} = v^2 + 2v/x \hfill \\
\hfill \\
\int {\frac{{dv}}
{{v(v + 2v)}} = \int {\frac{{dx}}
{x} = > \int {(\frac{1}
{v} - \frac{1}
{{1 + v}})dv = \int {\frac{{dx}}
{x}} } } } \hfill \\
\end{gathered}
$

$
\ln |v/2 + v| = \ln |x| + A
$

......

I need to know how to get to this....

$
\frac{x}
{{A - \ln |x|}} - x
$

2. Originally Posted by dankelly07
Just need a bit of help finishing this one, not sure if I'm going about this correctly...

$
\begin{gathered}
\frac{{dy}}
{{dx}} = \frac{{y^2 + 3xy + x^2 }}
{{x^2 }} \hfill \\
\frac{{dy}}
{{dx}} = \frac{{v^2 + 3v}}
{x} - \frac{v}
{x} \hfill \\
\frac{{dy}}
{{dx}} = v^2 + 2v/x \hfill \\
\hfill \\
\int {\frac{{dv}}
{{v(v + 2v)}} = \int {\frac{{dx}}
{x} = > \int {(\frac{1}
{v} - \frac{1}
{{1 + v}})dv = \int {\frac{{dx}}
{x}} } } } \hfill \\
\end{gathered}
$

$
\ln |v/2 + v| = \ln |x| + A
$

......

I need to know how to get to this....

$
\frac{x}
{{A - \ln |x|}} - x
$
$\frac{dy}{dx} = \frac{y^2 + 3xy + x^2 }{x^2}$

With $y = vx$ substitution:

$v + x\frac{dv}{dx} = v^2 + 3v + 1$
$x\frac{dv}{dx} = v^2 + 2v + 1 = (v+1)^2$
$\int \frac{dv}{(v+1)^2} = \int \frac{dx}{x}$
$-\frac1{v+1} = \ln C|x|$
$v = -\frac1{\ln C|x|} - 1 \Rightarrow y = -\frac{x}{\ln C|x|} - x$

So the solution is $y = \frac{x}{A - \ln|x|} - x$ where $A = - \ln C$, where A is as arbitrary as C is.*

*- These are Mr.Fantastic's words... I like it.