Nonlinear DE similar to a Bernoulli equation

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September 4th 2012, 10:49 AM
bradbrad

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.

Brad.
September 4th 2012, 06:23 PM
Prove It

Re: Nonlinear DE similar to a Bernoulli equation

Quote:

Originally Posted by bradbrad

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.

Brad.

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?
September 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.

Hi brandbrad !

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.
http://www.google.fr/url?sa=t&rct=j&q="an abel ordinary differential equation"%2C0002059v3&source=web&cd=1&cad=rja&ved= 0CCMQFjAA&url=http%3A%2F%2Farxiv.org%2Fpdf%2Fmath% 2F0002059&ei=dO1GUPfQGejZ0QXot4C4AQ&usg=AFQjCNHVF0 JbDhUtCSYorBYWHFfyz6JbfQ
In case of n>3, analytical solving in not possible in general, except in cases of particular forms of functions f(x), g(x), h(x). In practice, numerical solving is requiered.
September 8th 2012, 07:47 AM
bradbrad

Re: Nonlinear DE similar to a Bernoulli equation

Hi Prove It and JJ! Many thanks for your input, much appreciated =)

As it turns out, this is Chini's equation, and as JJ mentions is only generally solvable for particular n.

I think I'm just going to linearize it and settle for an approximate solution.

Many thanks guys!