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]

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