# Reducing non-linear ODE to Linear

• Feb 13th 2013, 08:49 AM
acid_rainbow
Reducing non-linear ODE to Linear
Hi all,

I'm struggling with the following problem.

Consider basic non linear differential equation:

dx/dt + a(t)x = b(t)xn

I understand that I can define a new variable y = x1-n

However I am unsure how to make the general substituion to get to the solution:

dy/dt + (1-n)a(t)y = (1-n)b(t)

I would really appreciate it if someone could point me in the right direction and provide me with the steps.

I have tried differentiating it and substituting in for x dot but I still have x terms floating around.

Thank you for the help!
• Feb 13th 2013, 09:17 AM
a tutor
Re: Reducing non-linear ODE to Linear
Differentiate $\displaystyle y=x^{1-n}$ with respect to t so that $\displaystyle \frac{dy}{dt}=\frac{1-n}{x^n}\frac{dx}{dt}$ or $\displaystyle \frac{dx}{dt}=\frac{x^n}{1-n}\frac{dy}{dt}$.

Sub it in along with the other bit then multiply through by 1-n and divide by x^n.
• Feb 13th 2013, 09:51 AM
acid_rainbow
Re: Reducing non-linear ODE to Linear
Thank's it just clicked. Much appreciated :)