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There is a symbolic equation system like this and wonder if any one could help me out by solving this equation system symbolically:

- 4 equations with 4 unknown variables [u ̂ , v ̂, w ̂, p ̂ ]. (u^, v^, w^, p^ are function of "r", namely, u(r), v(r), w(r), p(r))
- All other symbolic parameters are known variables, i.e., A, r, m, alpha, U(r), V(r), W(r) ).

Here are the equations:

Objective :

- Combine the 4 equations into 1 single equation in terms of u^ only, namely, cancel out v ̂, w ̂, p ̂ .
- Find expressions of v ̂, w ̂, p ̂ in terms of u^ : i.e., find expression of v^ in termed of u^, v^=f (u^). Same for expressions of w^ and p^, in terms of u^.

Here is a short program I made and I know it is not even close to the answer:

clc

clear

syms U(r) V(r) W(r) u(r) v(r) w(r) p(r) A r m alpha

% f(u, v, w, p)

eqn1=A*u+diff(U,r)*v+alpha*p==0; % f(u, v, p)=0

eqn2=-A*v-2*W/r*w+diff(p,r)==0; % f( v, w, p)=0

eqn3=(diff(W,r)+W/r)*v+A*w+m/r*p==0; % f( v, w, p)=0

eqn4=alpha*u+v/r+diff(v,r)+m/r*w==0; % f(u, v, w )=0

solve([eqn1,eqn2,eqn3,eqn4],[v w p])

The above program doesn solve for "v" w" "p" in terms of "u". I wondered is there anyway to do it? I really appreciate your time and help.

Walter Roberson
on 26 Feb 2021

That is a series of ODE. You will not be able to solve it with solve(). You could try it with

dsolve([eqn1, eqn2, eqn3, eqn4])

but the reality is that you will get back emptiness.

There is a trivial solution with u, v, r, and p all being constant 0s. Other than that there does not appear to be any closed form solutions.

Any time you have an unknown function involved U(r), V(r), W(r) then you are not going to get a closed form solution except possibly trivial solutions (like the constant zeros.)

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