How to solve 2nd order differential equations while variables are coupled?

(w-1)*U1r-d(U1i)/dx+L*U2r-m*d^2(U1r)/dx^2=0;

(w+1)*U1i+d(U1r)/dx+L*U2i+m*d^2(U1i)/dx^2=0;

(w-1)*U2r-d(U2i)/dx+L*U1r-m*d^2(U2r)/dx^2=0;

(w+1)*U2i+d(U2r)/dx+L*U1i+m*d^2(U2i)/dx^2=0;

How to solve U1r, U1i, U2r, and U2i by using MATLAB symlbolic tool ?

Thanks in advance.

Answer 1

[Birdman]

Try the following code:

syms w U1r(x) U1i(x) U2i(x) U2r(x) L m d
eq1=(w-1)*U1r-diff(U1i)+L*U2r-m*diff(U1r,2)==0; 
eq2=(w+1)*U1i+diff(U1r)+L*U2i+m*diff(U1i,2)==0; 
eq3=(w-1)*U2r-diff(U2i)+L*U1r-m*diff(U2r,2)==0;
eq4=(w+1)*U2i+diff(U2r)+L*U1i+m*diff(U2i,2)==0;
solx=dsolve([eq1 eq2 eq3 eq4]);
%%solutions
U1r(x)=solx.U1r;
U1i(x)=solx.U1i;
U2r(x)=solx.U2r;
U2i(x)=solx.U2i;

[Md Bellal Hossain]

Thanks for your answer.

But, I found the solution is too large. How can I smiplify the solution?

[Birdman]

You may use

simplify

command.

[Md Bellal Hossain]

Thanks for your reply. I know it but still it can't simply that large solution.

[Birdman]

Maybe you can insert numerical values for L, m and d and you can both simplify the solution and obtain numerical results at the same time :)

[Md Bellal Hossain]

Thanks a lot. Have a great day.

[Md Bellal Hossain]

(w-1)*U1r-d(U1i)/dx+L*U2r-m*d^2(U1r)/dx^2=0;

(w+1)*U1i+d(U1r)/dx+L*U2i+m*d^2(U1i)/dx^2=0;

(w-1)*U2r-d(U2i)/dx+L*U1r=0;

(w+1)*U2i+d(U2r)/dx+L*U1i=0;

How to solve analytically U1r, U1i, U2r, and U2i by using MATLAB symlbolic tool ?

Thanks in advance.