differential equations-help!!!!!!!

**mbempeni** · February 6th 2009, 12:08 PM

Please help - question attached.

**Soroban** · February 6th 2009, 12:35 PM

Hello, mbempeni!

The first one is homegeneous . . .

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

Divide top and bottom of the fraction by $x^3\!:\quad\frac{dy}{dx} \;=\;\frac{1 - 2\left(\dfrac{y}{x}\right) - \left(\dfrac{y}{x}\right)^3}{1 - \left(\dfrac{y}{x}\right)^2}$

Let $v \,=\,\frac{y}{x}\quad\Rightarrow\quad y \,=\,vx\quad\Rightarrow\quad \frac{dy}{dx} \:=\:v + x\frac{dv}{dx}$

Substitute: . $v + x\frac{dv}{dx} \;=\;\frac{1-2v-v^3}{1-v^2}$

Then: . $x\frac{dv}{dx} \;=\;\frac{1-2v-v^3}{1-v^2} - v \quad\Rightarrow\quad x\frac{dv}{dx}\;=\;\frac{1-3v}{1-v^2}$

Separate variables: . $\frac{v^2-1}{3v-1}\,dv \;=\;\frac{dx}{x}$

Can you finish it now?

**mbempeni** · February 7th 2009, 02:37 AM

Originally Posted by Soroban

Hello, mbempeni!

The first one is homegeneous . . .

Divide top and bottom of the fraction by $x^3\!:\quad\frac{dy}{dx} \;=\;\frac{1 - 2\left(\dfrac{y}{x}\right) - \left(\dfrac{y}{x}\right)^3}{1 - \left(\dfrac{y}{x}\right)^2}$

Let $v \,=\,\frac{y}{x}\quad\Rightarrow\quad y \,=\,vx\quad\Rightarrow\quad \frac{dy}{dx} \:=\:v + x\frac{dv}{dx}$

Substitute: . $v + x\frac{dv}{dx} \;=\;\frac{1-2v-v^3}{1-v^2}$

Then: . $x\frac{dv}{dx} \;=\;\frac{1-2v-v^3}{1-v^2} - v \quad\Rightarrow\quad x\frac{dv}{dx}\;=\;\frac{1-3v}{1-v^2}$

Separate variables: . $\frac{v^2-1}{3v-1}\,dv \;=\;\frac{dx}{x}$

Can you finish it now?

Can you finish it?It much more difficult than I thought.Please..

**mr fantastic** · February 7th 2009, 02:47 AM

Originally Posted by mbempeni

Can you finish it?It much more difficult than I thought.Please..

Surely you can integrate the right hand side.

As for the left hand side, note that $\frac{v^2 - 1}{3v - 1} = \frac{v}{3} + \frac{1}{9} - \frac{8}{9} \left( \frac{1}{3v - 1}\right)$ .

If you're studying differential equations then it's expected that you can apply the various techniques of integration that will be required and which you have no doubt been taught. It might be wise to go back and revise them.

Thread: differential equations-help!!!!!!!

LinkBack

Thread Tools

Display

differential equations-help!!!!!!!

Similar Math Help Forum Discussions

'Differential' in differential equations

Differential Equations-Application of 1st order differential equations 1

Differential Equations:Linear first Order Equations

First Order Differential Equations: Separable Equations and Applications

Differential equations

Search Tags