# Help with using the Runge-Kutta 4th order method on a system of 2 first order ODE's.

The original ODE I had was $$\frac{d^2y}{dx^2}+\frac{dy}{dx}-6y=0$$ with $y(0)=3$ and $y'(0)=1$. Now I can solve this by hand and obtain that $y(1) = 14.82789927$. However I wish to use the 4th order Runge-Kutta method, so I have the system:

$$\left\{\begin{array}{l} \frac{dy}{dx} = z \\ \frac{dz}{dx} = 6y - z \end{array}\right.$$ With $y(0)=3$ and $z(0)=1$.

Now I know that for two general 1st order ODE's $$\frac{dy}{dx} = f(x,y,z) \\ \frac{dz}{dx}=g(x,y,z)$$ The 4th order Runge-Kutta formula's for a system of 2 ODE's are: $$y_{i+1}=y_i + \frac{1}{6}(k_0+2k_1+2k_2+k_3) \\ z_{i+1}=z_i + \frac{1}{6}(l_0+2l_1+2l_2+l_3)$$ Where $$k_0 = hf(x_i,y_i,z_i) \\ k_1 = hf(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_i+\frac{1}{2}l_0) \\ k_2 = hf(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1,z_i+\frac{1}{2}l_1) \\ k_3 = hf(x_i+h,y_i+k_2,z_i+l_2)$$ and $$l_0 = hg(x_i,y_i,z_i) \\ l_1 = hg(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_i+\frac{1}{2}l_0) \\ l_2 = hg(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1,z_i+\frac{1}{2}l_1) \\ l_3 = hg(x_i+h,y_i+k_2,z_i+l_2)$$

My problem is I am struggling to apply this method to my system of ODE's so that I can program a method that can solve any system of 2 first order ODE's using the formulas above, I would like for someone to please run through one step of the method, so I can understand it better.

• For reference, see this answer on SO. Commented Mar 24, 2018 at 15:22

I will outline the process and you can fill in the calculations.

We have our system as:

$$\left\{\begin{array}{l} \frac{dy}{dx} = z \\ \frac{dz}{dx} = 6y - z \end{array}\right.$$ With $y(0)=3$ and $z(0)=1$.

We must do the calculations in a certain order as there are dependencies between the numerical calculations. This order is:

• $k_0 = hf(x_i,y_i,z_i)$
• $l_0 = hg(x_i,y_i,z_i)$

• $k_1 = hf(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_i+\frac{1}{2}l_0)$

• $l_1 = hg(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_i+\frac{1}{2}l_0)$

• $k_2 = hf(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1,z_i+\frac{1}{2}l_1)$

• $l_2 = hg(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1,z_i+\frac{1}{2}l_1)$

• $k_3 = hf(x_i+h,y_i+k_2,z_i+l_2)$

• $l_3 = hg(x_i+h,y_i+k_2,z_i+l_2)$

• $y_{i+1}=y_i + \frac{1}{6}(k_0+2k_1+2k_2+k_3)$

• $z_{i+1}=z_i + \frac{1}{6}(l_0+2l_1+2l_2+l_3)$

We typically need some inputs for the algorithm:

• A range that we want to do the calculations over: $a \le t \le b$, lets use $a = 0, b = 1$.
• The number of steps $N$, say $N = 10$.
• The steps size $h = \dfrac{b-a}{N} = \dfrac{1}{10}$

The system we are solving is:

$$\frac{dy}{dx} = f(x,y,z) = z \\ \frac{dz}{dx}=g(x,y,z) = 6y - z$$

Doing the calculations using the above order for the first time step $i= 0, t_0 = 0 = x_0$, yields:

• $k_0 = hf(x_0,y_0,z_0) = \dfrac{1}{10}(z_0) = \dfrac{1}{10}(1) = \dfrac{1}{10}$
• $l_0 = hg(x_0,y_0,z_0) = \dfrac{1}{10}(6y_0 - z_0) = \dfrac{1}{10}(6 \times 3 - 1) = \dfrac{1}{10}(17)$

• $k_1 = hf(x_0+\frac{1}{2}h,y_0+\frac{1}{2}k_0,z_0+\frac{1}{2}l_0) = \dfrac{1}{10}(1 + \dfrac{1}{2}\dfrac{1}{10}(17)) ~~$(You please continue the calcs.)

• $l_1 = hg(x_0+\frac{1}{2}h,y_i+\frac{1}{2}k_0,z_0+\frac{1}{2}l_0)$

• $k_2 = hf(x_0+\frac{1}{2}h,y_0+\frac{1}{2}k_1,z_0+\frac{1}{2}l_1)$

• $l_2 = hg(x_0+\frac{1}{2}h,y_0+\frac{1}{2}k_1,z_0+\frac{1}{2}l_1)$

• $k_3 = hf(x_0+h,y_0+k_2,z_0+l_2)$

• $l_3 = hg(x_0+h,y_0+k_2,z_0+l_2)$

• $y_{1}=y_0 + \frac{1}{6}(k_0+2k_1+2k_2+k_3)$

• $z_{1}=z_0 + \frac{1}{6}(l_0+2l_1+2l_2+l_3)$

You now have $x_1$ and $z_1$ which you need for the next time step after all of the intermediate (in order again). Now, we move on to the next time step $i = 1, t_1 = t_0 + h = \dfrac{1}{10} = x_1$, so we have:

• $k_0 = hf(x_1,y_1,z_1) = \dfrac{1}{10}(z_1)$
• $l_0 = hg(x_1,y_1,z_1) = \dfrac{1}{10}(6y_1 - z_1)$

• $k_1 = hf(x_1+\frac{1}{2}h,y_1+\frac{1}{2}k_0,z_1+\frac{1}{2}l_0)$

• $l_1 = hg(x_1+\frac{1}{2}h,y_1+\frac{1}{2}k_0,z_1+\frac{1}{2}l_0)$

• $k_2 = hf(x_1+\frac{1}{2}h,y_1+\frac{1}{2}k_1,z_1+\frac{1}{2}l_1)$

• $l_2 = hg(x_1+\frac{1}{2}h,y_1+\frac{1}{2}k_1,z_1+\frac{1}{2}l_1)$

• $k_3 = hf(x_1+h,y_1+k_2,z_1+l_2)$

• $l_3 = hg(x_1+h,y_1+k_2,z_1+l_2)$

• $y_{2}=y_1 + \frac{1}{6}(k_0+2k_1+2k_2+k_3)$

• $z_{2}=z_1 + \frac{1}{6}(l_0+2l_1+2l_2+l_3)$

Continue this for $10$ time steps. Your final result should match closely (assuming the numerical algorithm is stable for this problem) to the exact solution. You will compare $z_{10}$ to the exact result. The exact solution is:

$$y(x) = e^{-3 x}+2 e^{2 x}$$

If we find $y(1) = \dfrac{1}{e^3} + 2 e^2 = 14.8278992662291643974401973...$.

• Thanks for your thorough response, seeing the start of it I now understand it better. Thanks! Commented Mar 21, 2014 at 15:49
• Recall, in your new system, the first equation $y' = z$ is just a dummy variable in order to use RK4 methods. Regards Commented Mar 21, 2014 at 15:53
• @Michael: Also, you will clearly see when you calculate $y_i, z_i$, which is the correct final result. Commented Mar 21, 2014 at 16:02
• Late reply but, is $y_i$ or $z_i$ the solution to the original ODE? Comparing values it seems like the solution is given by $y_i$, but I'm not sure. Commented May 4, 2016 at 15:47
• @ErikVesterlund if I got this right, then z would be the solution for the derivative of y and y is the solution to the original ODE Commented Jan 22, 2017 at 16:11

Although this answer contains the same content as Amzoti's answer, I think it's worthwhile to see it another way.

In general consider if you had $$m$$ first-order ODE's (after appropriate decomposition). The system looks like

\begin{align*} \frac{d y_1}{d x} &= f_1(x, y_1, \ldots, y_m) \\ \frac{d y_2}{d x} &= f_2(x, y_1, \ldots, y_m) \\ &\,\,\,\vdots\\ \frac{d y_m}{d x} &= f_m(x, y_1, \ldots, y_m) \\ \end{align*}

Define the vectors $$\vec{Y} = (y_1, \ldots, y_m)$$ and $$\vec{f} = (f_1, \ldots, f_m)$$, then we can write the system as

$$\frac{d}{dx} \vec{Y} = \vec{f}(x,\vec{Y})$$

Now we can generalize the RK method by defining \begin{align*} \vec{k}_1 &= h\vec{f}\left(x_n,\vec{Y}(x_n)\right)\\ \vec{k}_2 &= h\vec{f}\left(x_n + \tfrac{1}{2}h,\vec{Y}(x_n) + \tfrac{1}{2}\vec{k}_1\right)\\ \vec{k}_3 &= h\vec{f}\left(x_n + \tfrac{1}{2}h,\vec{Y}(x_n) + \tfrac{1}{2}\vec{k}_2\right)\\ \vec{k}_4 &= h\vec{f}\left(x_n + h, \vec{Y}(x_n) + \vec{k}_3\right) \end{align*}

and the solutions are then given by $$\vec{Y}(x_{n+1}) = \vec{Y}(x_n) + \tfrac{1}{6}\left(\vec{k}_1 + 2\vec{k}_2 + 2\vec{k}_3 + \vec{k}_4\right)$$

with $$m$$ initial conditions specified by $$\vec{Y}(x_0)$$. When writing a code to implement this one can simply use arrays, and write a function to compute $$\vec{f}(x,\vec{Y})$$

For the example provided, we have $$\vec{Y} = (y,z)$$ and $$\vec{f} = (z, 6y-z)$$. Here's an example in Fortran90:

program RK4
implicit none
integer , parameter :: dp = kind(0.d0)
integer , parameter :: m = 2 ! order of ODE
real(dp) :: Y(m)
real(dp) :: a, b, x, h
integer :: N, i

! Number of steps
N = 10

! initial x
a = 0
x = a

! final x
b = 1

! step size
h = (b-a)/N

! initial conditions
Y(1) = 3 ! y(0)
Y(2) = 1 ! y'(0)

! iterate N times
do i = 1,N
Y = iterate(x, Y)
x = x + h
end do

print*, Y

contains

! function f computes the vector f

function f(x, Yvec) result (fvec)
real(dp) :: x
real(dp) :: Yvec(m), fvec(m)

fvec(1) = Yvec(2) !z
fvec(2) = 6*Yvec(1) - Yvec(2) !6y-z

end function

! function iterate computes Y(t_n+1)

function iterate(x, Y_n) result (Y_nplus1)
real(dp) :: x
real(dp) :: Y_n(m), Y_nplus1(m)
real(dp) :: k1(m), k2(m), k3(m), k4(m)

k1 = h*f(x, Y_n)
k2 = h*f(x + h/2, Y_n + k1/2)
k3 = h*f(x + h/2, Y_n + k2/2)
k4 = h*f(x + h, Y_n + k3)

Y_nplus1 = Y_n + (k1 + 2*k2 + 2*k3 + k4)/6

end function

end program


This can be applied to any set of $$m$$ first order ODE's, just change m in the code and change the function f to whatever is appropriate for the system of interest. Running this code as-is yields

$$14.827578509968953 \qquad 29.406156886687729$$

The first value is $$y(1)$$, the second $$z(1)$$, correct to the third decimal point with only ten steps.

• you should use $x_{n}$ instead of $t_{n}$ Commented Nov 3, 2018 at 20:39
• Good catch, fixed it
– Kai
Commented Nov 4, 2018 at 0:20
• Fantastic answer @Kai +1. Would give +50 if possible! Many people struggle with systems of ODE's and RK methods. I have a question though regarding your Fortran implementation. If you wanted to be fancy you could write your $k_i$'s using a for loop correct? Essentially placing them in an array? So you would have an array $k(i,n)$ where i was the number of stages and n was the dimension of your state vector? Are you aware of any documentation that does this in Fortran? I am writing something similar at the minute and am a bit stumped!! Commented Feb 10, 2019 at 0:03
• I have noticed that there is no explicit dependence of function f on variable x. Is that because of the substitution we used to reduce the order of derivative? Will this dependence on x be automatically picked up by the algorithm through other variables? Commented Jun 10, 2022 at 10:31
• @MadPhysicist the ODE in the original question didn't have explicit $x$ dependence, it's only implicit in $y(x)$ and $z(x)$. You could have explicit $x$ dependence of course
– Kai
Commented Jun 10, 2022 at 14:49

A Matlab implementation is given below:

% It calculates ODE using Runge-Kutta 4th order method
% Author Ido Schwartz
% Originally available form: http://www.mathworks.com/matlabcentral/fileexchange/29851-runge-kutta-4th-order-ode/content/Runge_Kutta_4.m
% Edited by Amin A. Mohammed, for 2 ODEs(April 2016)

clc;                                               % Clears the screen
clear all;

h=0.1;                                             % step size
x = 0:h:1;                                         % Calculates upto y(1)
y = zeros(1,length(x));
z = zeros(1,length(x));
y(1) = 3;                                          % initial condition
z(1) = 1;                                          % initial condition
% F_xy = @(t,r) 3.*exp(-t)-0.4*r;                  % change the function as you desire
F_xyz = @(x,y,z) z;                                  % change the function as you desire
G_xyz = @(x,y,z) 6*y-z;

for i=1:(length(x)-1)                              % calculation loop
k_1 = F_xyz(x(i),y(i),z(i));
L_1 = G_xyz(x(i),y(i),z(i));
k_2 = F_xyz(x(i)+0.5*h,y(i)+0.5*h*k_1,z(i)+0.5*h*L_1);
L_2 = G_xyz(x(i)+0.5*h,y(i)+0.5*h*k_1,z(i)+0.5*h*L_1);
k_3 = F_xyz((x(i)+0.5*h),(y(i)+0.5*h*k_2),(z(i)+0.5*h*L_2));
L_3 = G_xyz((x(i)+0.5*h),(y(i)+0.5*h*k_2),(z(i)+0.5*h*L_2));
k_4 = F_xyz((x(i)+h),(y(i)+k_3*h),(z(i)+L_3*h)); % Corrected
L_4 = G_xyz((x(i)+h),(y(i)+k_3*h),(z(i)+L_3*h));

y(i+1) = y(i) + (1/6)*(k_1+2*k_2+2*k_3+k_4)*h;  % main equation
z(i+1) = z(i) + (1/6)*(L_1+2*L_2+2*L_3+L_4)*h;  % main equation

end


A Fortran code shown below:

produces the following result

!Runge-Kutta Fourth Order Method

!For 2nd Order Differentiation Equation

!First you have to define the function

F(x,y,z) = z !dy/dx

G(x,y,z) = 6*y-z !dz/dx = d2y/dx2

INTEGER :: n,i

REAL :: k1,l1,k2,l2,k3,l3,k4,l4    !Most Important

Write (*,*) "Given Equation '(y2)-6(y1)+(y0)=0'"

Write (*,*) "Xo=0, Yo=3, Zo=Y'o=1, Xn=1, n=?"

Xo=0    !Given Condition

Yo=3    !Given Condition

Zo=1    !Given Condition

Xn=1    !Given Condition

read (*,*) n    !n=number of Intercept

h=(Xn-Xo)/n

do i=1,n    !you have to do the Calculation 'n' times

k1 = h*F(Xo,Yo,Zo)

l1 = h*G(Xo,Yo,Zo)

k2 = h*F(Xo+h/2,Yo+k1/2,Zo+l1/2)

l2 = h*G(Xo+h/2,Yo+k1/2,Zo+l1/2)

k3 = h*F(Xo+h/2,Yo+k2/2,Zo+l2/2)

l3 = h*G(Xo+h/2,Yo+k2/2,Zo+l2/2)

k4 = h*F(Xo+h,Yo+k3,Zo+l3)

l4 = h*G(Xo+h,Yo+k3,Zo+l3)

!Sum Up

Yn = Yo+(k1+2*k2+2*k3+k4)/6

Zn = Zo+(l1+2*l2+2*l3+l4)/6

!Operation for Next calculation

Xo=Xo+h     !(+h) than previous Term

Yo=Yn    !Now Yn becomes Yo

Zo=Zn    !Now Zn becomes Zo

End Do

Write (*,*) "Xn,Yn =",Xo,Yo

Stop

End

• Can your clarify your question? Commented Jan 31, 2019 at 19:39

A Rust implementation is below:

const M: usize = 2; // Size of the system

fn main() {
let mut y: [f64; M] = [0.0; M];
let mut a: f64 = 0.0;
let mut x: f64 = a;
let mut b: f64 = 0.0;
let n: usize = 10;
let h: f64 = (b - a) / n as f64;

// Initial conditions
y[0] = 3.0; // y(0)
y[1] = 1.0; // y'(0)

for _i in 0..n {
y = rk4_step(x, y, h, &derivatives);
x += h;
}

println!("{:?}", y);
}

// Define the derivatives for the system of equations
fn derivatives(x: f64, y: [f64; M]) -> [f64; M] {
let mut dy_dx: [f64; M] = [0.0; M];

dy_dx[0] = y[1]; // dy/dx = z
dy_dx[1] = 6.0 * y[0] - y[1]; // dz/dx = 6y - z

dy_dx
}

// Perform one step of RK4
fn rk4_step(x: f64, y_n: [f64; M], h: f64, f: &dyn Fn(f64, [f64; M]) -> [f64; M]) -> [f64; M] {
let mut k1: [f64; M] = [0.0; M];
let mut k2: [f64; M] = [0.0; M];
let mut k3: [f64; M] = [0.0; M];
let mut k4: [f64; M] = [0.0; M];
let mut y_n_plus_1: [f64; M] = [0.0; M];

k1 = scale_vec(&f(x, y_n), h);
k2 = scale_vec(&f(x + h / 2.0, add_vec(&y_n, &scale_vec(&k1, 0.5))), h);
k3 = scale_vec(&f(x + h / 2.0, add_vec(&y_n, &scale_vec(&k2, 0.5))), h);
k4 = scale_vec(&f(x + h, add_vec(&y_n, &k3)), h);

&y_n,
&scale_vec(
1.0 / 6.0,
),
);

y_n_plus_1
}

// Helper function: element-wise vector addition
fn add_vec(a: &[f64; M], b: &[f64; M]) -> [f64; M] {
let mut result: [f64; M] = [0.0; M];
for i in 0..M {
result[i] = a[i] + b[i];
}
result
}

// Helper function: element-wise scalar multiplication of a vector
fn scale_vec(a: &[f64; M], scalar: f64) -> [f64; M] {
let mut result: [f64; M] = [0.0; M];
for i in 0..M {
result[i] = a[i] * scalar;
}
result
}