# Nonlinear DE similar to a Bernoulli equation

• Sep 4th 2012, 10:49 AM
Nonlinear DE similar to a Bernoulli equation
Hi all,

I've got a nonlinear differential equation of the general form

y' + f(x)y + g(x) = h(x)(y^n)

to solve.

For g(x) = 0 this is your bog-standard Bernoulli equation. I've been trying to think of a way to solve it but haven't managed so far.

Any ideas would be appreciated.

Many thanks.

• Sep 4th 2012, 06:23 PM
Prove It
Re: Nonlinear DE similar to a Bernoulli equation
Quote:

Hi all,

I've got a nonlinear differential equation of the general form

y' + f(x)y + g(x) = h(x)(y^n)

to solve.

For g(x) = 0 this is your bog-standard Bernoulli equation. I've been trying to think of a way to solve it but haven't managed so far.

Any ideas would be appreciated.

Many thanks.

I am not sure if this is correct, but this would be my thinking.

When we want to solve a second order linear DE, the solution is the sum of the the homogeneous and the non-homogeneous solutions.

Since we can write this DE as \displaystyle \begin{align*} y' + f(x)\,y = h(x)\,y^n - g(x) \end{align*}, I would expect you could use a similar process here, where the solution would be equal to the sum of the solutions of \displaystyle \begin{align*} y' + f(x)\,y = 0 \end{align*} (homogeneous case), \displaystyle \begin{align*} y' + f(x)\,y = -g(x) \end{align*} (first order linear case) and \displaystyle \begin{align*} y' + f(x)\,y = h(x)\,y^n \end{align*} (Bernoulli case).

Is there anything wrong with my thinking?
• Sep 4th 2012, 10:40 PM
JJacquelin
Re: Nonlinear DE similar to a Bernoulli equation
Hi Prove It !

In the case of NON-linear ODE, the solution is NOT the sum of the the homogeneous and the non-homogeneous solutions.

As already said, "For g(x) = 0 this is your bog-standard Bernoulli equation".
More generally, if g(x) exists, the equation can be analytically solved only in some cases.
Case n=2 : This is a Riccati equation, which method of solving is known ( transformation into a linear second order ODE)
Case n=3 : This an Abel equation, which can be analytically solved only in a few particular cases. For example see :
"An Abel ordinary differential equation class generalizing known integrable classes", E.S. Cheb-Terrab 1,2,and A.D. Roche.