Seperable Intial-Value Problem

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January 30th 2011, 09:38 AM
JJ007

Seperable Intial-Value Problem

$(x^2+3y^2)dx + (2xy)dy = 0$
$y(2)=6$

so this homogeneous eqn. needs to be transformed into a separable eqn. right?
$y=vx , v=\frac{y}{x}$

$\frac{dy}{dx}=\frac{x^2+3y^2}{2xy}$ or is it $\frac{x^2+3y^2}{2x}dx - \frac{y}{1}dy$ = How can I plug in $\frac{y}{x}$ here ?

$\frac {dy}{dx}= v + x\frac{dv}{dx}$

Not sure how to solve this one.
January 30th 2011, 09:53 AM
TheEmptySet

Quote:

Originally Posted by JJ007

$(x^2+3y^2)dx + (2xy)dy = 0$
$y(2)=6$

so this homogeneous eqn. needs to be transformed into a separable eqn. right?
$y=vx , v=\frac{y}{x}$

$\frac{dy}{dx}=\frac{x^2+3y^2}{2xy}$ = How can I plug in $\frac{y}{x}$ here ?

$\frac {dy}{dx}= v + x\frac{dv}{dx}$

Not sure how to solve this one.

Multiply the numerator and denominator by

$\frac{1}{x^2}$

This gives

$\displaystyle \frac{1+3\left( \frac{y}{x}\right)^2}{2\left( \frac{y}{x}\right)}$
February 4th 2011, 05:00 AM
dwsmith

Quote:

Originally Posted by JJ007

$\frac {dy}{dx}= v + x\frac{dv}{dx}$

You should write the above as follows:

$y=vx\Rightarrow dy=xdv+vdx$

You would then plug this into dy of the DE.

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