# Adapt RK4 ODE Solver from first order to 2nd order

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I have a Matlab function for doing Runge-Kutta4k approximation for first-order ODE's, and I want to adapt it to work for second-order ODE's. This is what I have for first order RK4 ( and it's not working x) ) Would anyone be able to help me find a solution ?

function yt = RK4_2ndOrder(dxdt,y0,h,tfinal)

yt = zeros(2,length(h:tfinal)); %Memory Allocation

yt(:,1) = y0; %Initial condition

i = 2;

for t = h : h : tfinal %RK4 loop

k1 = dxdt(t-h,yt(:,i-1));

k2 = dxdt(t - (0.5*h), yt(:,i-1) + 0.5*k1*h);

k3 = dxdt(t - (0.5*h), yt(:,i-1) + 0.5*k1*h);

k4 = dxdt(t, yt(:,i-1) + (k3 * h));

yt(:,i) = yt(:,i-1) + (1/6 * (k1 + 2*k2 + 2*k3 + k4)* h);

i = i + 1;

end

end

This is my main

clc;

clear;

h = 0.01; %Time step

y0 = 1; %Initial condition dx1/dt = 1 m.s

tfinal = 20;

tarray = 0:h:tfinal;

ytRK4 = RK4(@dxdt,y0,h,tfinal);

plot(tarray,ytRK4_2ndOrder,'g');

And my function

function dx = dxdt(t,x)

m1 = 12000;

m2 = 10000;

m3 = 8000;

k1 = 3000;

k2 = 2400;

k3 = 1800;

dx = [ x(4) ; x(5) ; x(6) ; -k1/m1*x(1)+k2/m1*(x(2)-x(1)) ; k2/m2*(x(1)-x(2))+k3/m2(x(3)-x(2)) ; k3/m3*(x(2)-x(3)) ];

end

When I try to run my program, it says "Index exceeds the number of array elements" and "Error in dxdt (line 8)

dx = [x(4);x(5);x(6);-k1/m1*x(1)+k2/m1*(x(2)-x(1));k2/m2*(x(1)-x(2))+k3/m2(x(3)-x(2));k3/m3*(x(2)-x(3))];". As a beginner on Matlab, I don't really understand what that means...

Thanks in advance for the help !

##### 3 Comments

James Tursa
on 7 May 2021

### Accepted Answer

James Tursa
on 7 May 2021

Try this for starters

yt = zeros(numel(y0),length(h:tfinal)); %Memory Allocation

And then maybe this is what is intended for initial values:

y0 = [0;0;0;1;0;0]; %Initial condition dx1/dt = 1 m.s

And remember to fix this line for the k2:

k3 = dxdt(t - (0.5*h), yt(:,i-1) + 0.5*k2*h);

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