Solving System of Nonlinear Differential equations

I Need to solve the following system with ode45: (first order in R and second order in K)

R'(t)^2+a=b*K(t)^2*R(t)^(-1)

K''(t)+3*R(t)^2*R'(t)*K'(t)=c*R(t)^(-3)*K(t)^(-1)

with initial values:

R(0)=1

K(0)=3

K'(0)=0

I never used Matlab before, so would be great if someone could help me.

Answer 1

[Alan Stevens]

The following structure is what you need (notice that the second order ode is turned into two first order ones);

R0 = 1;
K0 = 3;
vK0 = 0;
Y0 = [R0 K0 vK0];
tspan = [0 10];  % Replace with your desired range
[t, Y] = ode45(@rate, tspan, Y0);

R = Y(:,1);
K = Y(:,2);

plot(t,R)
% etc.


function dYdt = rate(~,Y)

        a = 1; % Replace with your own value
        b = 1; % Ditto
        c = 1; % Ditto
        R = Y(1);
        K = Y(2);
        vK = Y(3);
                
        dRdt = sqrt( b*K^2/R - a );
        dKdt = vK;
        dvKdt = c/(R^3*K) - 3*R^2*dRdt*dKdt;
        
        
        dYdt = [dRdt;
                dKdt;
                dvKdt];
 
end

[Daniel Fröhlich]

Nice! Thanks a lot

[Cyrus Azizzadeh]

Hello

my equations system is like below. can I solve system like this in matlab?

dy(1) = 3*y(1) *t.^3 + y(2) + sin(dy(2)) + tan(dy(1)) ;

dy(2) = y(2) - y(1) + y(2).^4 + y(2).^4 + 3*t*dy(1) + t.^3*cos(y(1));