Number of solutions of a nonlinear differential equation.

[arroy_0205]

As far as I know, for an n-th order homogeneous linear differential equation, there are n number of linearly independent solutions and the general solution to the equation is a linear combination of them.
In the case of nth order homogeneous non-linear differential equation can it be shown that there are n number independent solutions? Can anybody tell me where I can find details of this? In case there are n number of independent solutions, I am not sure how to write the general solution. superposition principle will not hold. So what will be be the general solution? The degree of equation is one.

 

[HallsofIvy]

No, it can't. In fact, the whole idea of "independent solutions" or "independence" itself comes from Linear Algebra and only applies to linear equations.

 

[Defennder]

So how many solutions would there be to a nonlinear DE?

 

[HallsofIvy]

That depends very strongly on the specific non-linear equation!

 

[Defennder]

Sorry, I meant to say, for an "nth order non-linear DE".

 

[arroy_0205]

OK, so that means even for a second order nonlinear differential equation there may be 0/1/2 (or may be even more than 2, though not likely) solutions but there is no way to tell that (just by looking at the equation). On the contrary for any given linear 2nd order DE we know there are exactly 2 solutions.

 

[HallsofIvy]

We know that every solution can be written as a linear combination of two independent solutions. That's very different from saying "there are exactly 2 solutions".

 

[NoMoreExams]

Sorry, I meant to say, for an "nth order non-linear DE".

Some of those DEs might not even have a "nice" that is closed form solution. You might be interested in reading a few Dynamical System texts as they show how one can look at DEs by examining fixed points and their stability (as well as limit cycles, etc.) It builds quite nicely into Chaos Theory, Check out Strogatz's book for a nice, gentle introduction to the matter.

 

[Defennder]

I wasn't referring to the "nice" form of the solution. But can we even tell how many solutions there'll be for a nth order non-linear DE? And why doesn't the concept of linear independence apply here? Couldn't we just use the Wronskian to determine how many of them are linearly independent?

 

[HallsofIvy]

Linear indepence doesn't apply here because the equation is not linear!

Linear indepence is important in dealing with linear equations because of the fact that the set of solutions to a homogeneous n^th order differential equation for a vector space of dimension n. That is why any solution to an n^th order homogeneous differential equation can be written as a linear combination of n independent solutions: they form a basis for the vector space.

 

[matematikawan]

Does a concept of non-linearly combination of solutions makes any sense in trying to find a general solution for a non linear DE?

 

[zoki85]

Does a concept of non-linearly combination of solutions makes any sense in trying to find a general solution for a non linear DE?

Doesn't really,as far as I know
.

 

[smallphi]

Ok here is an example of my practise. I have a non-linear ODE of 3rd order (highest derivative is third). I've found a solution that depends on 3 constants which can be chosen at will. Does that mean I've found the most general solution of that ODE ?

Here I changed the original question from 'how many independent solutions of nonlinear ODE' to 'how many free constants there are in the solution of nonlinear ODE'. Can the new question be answered with certainty?

 

[matematikawan]

Doesn't really,as far as I know
.

I do came across a non-linear DE (Riccati equation) which has a nonlinear superposition formula.
It goes something this.

If y₁(x) , y₂(x) and y₃(x) are any three distict particular solutions of the Riccati equation and c is a constant, then the general solution is

$$\frac{cy_2(y_3-y_1)-y_1(y_3-y_2)}{c(y_3-y_1)-(y_3-y_2)}$$

I understand that this come naturally from Lie's Theorem which I don't understand.