Need keep in solove a system of differential equation

[hgyphy]

Need help in soloving a differential equation system

Hello all!


I met a second order differential equation system as followings. Would someone help in finding its solution?



U’’-a*V^2*U/r^2=0,

V’’+a*U^2*V/r^2=0,

a is a constant, u and v are function of r, that's u(r),v(r),


Great thanks

 

[chiro]

Hey hgyphy and welcome to the forums.

Have you tried using any kind separation of the variables?

Both terms only have two terms which means you can take one term to another side.

You should be able to integrate the LHS terms of each expression by parts to get an expression for the LHS in terms of V and V' and do the same for the other to get the terms for U and U'. Then you should be able to use those to get a mixed expression which you might have to use some kind of intergral/differentiation (ODE) technique.

As an example Integral of V''/V and U''/U can be done by parts to get the first part.

So this means you get:

V''/V = -aU^2/r^2. From this you should be able to get an expression for the integral of the LHS for V but you will still have an expression for the RHS.

Having said this you should get two expressions one for V and one for U (since they are in the same form). Once you do this you will probably have to use further tricks to actually solve explicitly V and U.

 

[hgyphy]

Hey hgyphy and welcome to the forums.

Have you tried using any kind separation of the variables?

Both terms only have two terms which means you can take one term to another side.

You should be able to integrate the LHS terms of each expression by parts to get an expression for the LHS in terms of V and V' and do the same for the other to get the terms for U and U'. Then you should be able to use those to get a mixed expression which you might have to use some kind of intergral/differentiation (ODE) technique.

As an example Integral of V''/V and U''/U can be done by parts to get the first part.

So this means you get:

V''/V = -aU^2/r^2. From this you should be able to get an expression for the integral of the LHS for V but you will still have an expression for the RHS.

Having said this you should get two expressions one for V and one for U (since they are in the same form). Once you do this you will probably have to use further tricks to actually solve explicitly V and U.

Thanks, Chiro! It seems still beyond my knowledge about differential equation. I'll appreciate if you or someone give me the whole solving process for thes headache equation.