How do I solve the ODE: x(1-x^2)+ky^2/y?

[Juggler123]

I need to solve the following ODE

\frac{dy}{dx}=\frac{x(1-x^2)+ky^2}{y}

I don't know what is the correct method to use though.

Any help would be brilliant, thanks.

 

[disregardthat]

Let u = y^2. Rearrange the equation as such: yy^{\prime}-ky^2 = x(1-x^2). Write the left hand side as a linear combination of u' and u.

 

[Juggler123]

O.k using

u=y^{2}

then does the equation become

\frac{du}{dx}-u=\frac{x(1-x^2)}{k}

I still don't know where to go from here (if this is even right!)

Am I missing something really easy here?

 

[dextercioby]

YOu can use the integrating factor method. If I'm not mistaking, the IF is e^(-x).

 

[sssssssssss]

wouldnt the integrating factor be e^-kx?

 

[dextercioby]

No, it's e^(-x).

 

[disregardthat]

It's e^(-kx) as you say, you made a mistake in your equation above.

 

[Redbelly98]

Moderator's note: thread moved to Homework & Coursework Questions.

Helpers please take note, the normal forum guidelines on giving homework help apply.