solve system of differential equation in matlab

Question

I am trying to solve a system of differential equations in Matlab.

dn/du=(-2*u*n-K*(n*u-(1+g)))/(1+u^2+K*u*(u-(1+g)/n))

dxi/du=(1-u^2)/(1+u^2+K*u*(u-(1+g)/n))

df/du=(2*u+K*u^2*(u-(1+g)/n))/(1+u^2+K*u*(u-(1+g)/n))

K and gamma are constants. I have the initial conditions like so:

n(0)=1, xi(0)=0, f(0)=0

First i tried to use 'dsolve'.

g=0.1;
K=3;

syms n(u) u 
n(u)=dsolve(diff(n,u)== (-2*u*n-K*(n*u-(1+g)))/(1+u^2+K*u*(u-(1+g)/n)),n(0)==1)

syms x(u) u n
x(u)= dsolve(diff(x,u)== (1-u^2)/(1+u^2+K*u*(u-(1+g)/n)),x(0)==0)

syms f(u) u n
f(u)=dsolve(diff(f,u)== (2*u+K*u^2*(u-(1+g)/n))/(1+u^2+K*u*(u-(1+g)/n)),f(0)==0)

from this i get that there is not explicit solution for the first equation and returned an [empty sym] as shown below.

Warning: Explicit solution could not be found.

In dsolve (line 201)

In Untitled (line 37)

n(u) = [ empty sym ]

x(u) = (1089*atan(33/(1600*n^2 - 1089)^(1/2) - (80*n*u)/(1600*n^2 - 1089)^(1/2)))/(160*n*(1600*n^2 - 1089)^(1/2)) - (35937*log(40*n*u^2 - 33*u + 10*n))/(2*(- 256000*n^3 + 174240*n)) - u/4 + (33*log(10*n)*(1600*n^2 - 1089)^(1/2) - 2178*atan(33/(1600*n^2 - 1089)^(1/2)) + 8000*n^2*atan(33/(1600*n^2 - 1089)^(1/2)))/(320*n*(1600*n^2 - 1089)^(1/2)) - (25*n*atan(33/(1600*n^2 - 1089)^(1/2) - (80*n*u)/(1600*n^2 - 1089)^(1/2)))/(1600*n^2 - 1089)^(1/2) + (26400*n^2*log(40*n*u^2 - 33*u + 10*n))/(- 256000*n^3 + 174240*n)

f(u) = (3*u^2)/8 - (33*u)/(160*n) - (log(40*n*u^2 - 33*u + 10*n)*(3200000*n^4 - 3920400*n^2 + 1185921))/(2*(- 10240000*n^4 + 6969600*n^2)) + (log(10*n)*(3200000*n^4 - 3920400*n^2 + 1185921))/(2*(- 10240000*n^4 + 6969600*n^2)) + (33*atan((1089*(2800*n^2 - 1089))/((1600*n^2 - 1089)^(1/2)*(92400*n^2 - 35937)))*(2800*n^2 - 1089))/(6400*n^2*(1600*n^2 - 1089)^(1/2)) - (33*atan((1089*(2800*n^2 - 1089))/((1600*n^2 - 1089)^(1/2)*(92400*n^2 - 35937)) - (2640*n u (2800*n^2 - 1089))/((1600*n^2 - 1089)^(1/2)*(92400*n^2 - 35937)))*(2800*n^2 - 1089))/(6400*n^2*(1600*n^2 - 1089)^(1/2))

Then i tried to use 'ode45'.

u=0:0.1:1;
K=3;
g=0.1;
dndu=@(u,n) (-2*u*n-K*(n*u-(1+g)))/(1+u^2+K*u*(u-(1+g)/n));
[u,n]=ode45(dndu, u, 1) % initial n=1

dxidu=@(u,xi) (1-u^2)/(1+u^2+K*u*(u-(1+g)/n));
[u,xi]=ode45(dxidu, u, 0); %initial xi=0

dfdu=@(u,f) (2*u+K*u^2*(u-(1+g)/n))/(1+u^2+K*u*(u-(1+g)/n));
[u,f]=ode45(dfdu, u, 0) %initial f=0

K and g are constants and u gets values for 0 to 1. When i run this the first equation is solved and got an answer for u and n but i get an error for the second equation that matrix dimension must agree as shown below.

u = 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000

n = 1.0000 1.3409 1.6243 1.7746 1.7945 1.7263 1.6119 1.4800 1.3470 1.2208 1.1048

Error using /
Matrix dimensions must agree.

Error in @(u,xi)(1-u^2)/(1+u^2+K u (u-(1+g)/n))

Error in odearguments (line 87)
f0 = feval(ode,t0,y0,args{:}); % ODE15I sets args{1} to yp0.

Error in ode45 (line 113)
[neq, tspan, ntspan, next, t0, tfinal, tdir, y0, f0, odeArgs, odeFcn, ...

Error in ode45method (line 24)
[u,xi]=ode45(dxidu, u, 0); %initial xi=0

Can anyone help me how to approach this type of problem? Thanks in advance!

Answer 1

You have a system of coupled differential equations, you need to solve it as a coupled system. One ODE function for a vector valued function with 3 components.

Answer 2

u1=0:0.01:1;
K=3;
g=0.1;
dydu=@(u,y)[(-2*u*y(1)-K*(y(1)*u-(1+g)))/(1+u^2+K*u*(u-(1+g)/y(1)))
        (1-u^2)/(1+u^2+K*u*(u-(1+g)/y(1)))
        (2*u+K*u^2*(u-(1+g)/y(1)))/(1+u^2+K*u*(u-(1+g)/y(1)))];
[U,Y]=ode45(dydu,u1,[1 0 0]);
n=Y(:,1);
xi=Y(:,2);
f=Y(:,3);