09 Mar 2026
Runge-Kutta Method for First-Order Differential Equations
Numerical solution of first-order differential equations using the fourth-order Runge-Kutta method.
Many physical laws give rates of change rather than direct formulas. For example, velocity gives the rate of change of position, and force determines the rate of change of momentum. A first-order ordinary differential equation has the form
\[\frac{dy}{dx}=f(x,y),\]with an initial condition
\[y(x_0)=y_0.\]The aim is to compute approximate values of $y$ at later points.
The initial condition gives the starting point of the curve. A numerical method then walks along the curve using a chosen step size.
Step form
Let
\[x_{n+1}=x_n+h.\]The numerical method estimates $y_{n+1}$ from $x_n$, $y_n$, and the step size $h$.
Euler’s method uses only the slope at the beginning of the interval. Runge-Kutta improves this idea by sampling slopes inside the interval as well.
Fourth-order Runge-Kutta method
The most commonly used form is
\[k_1=h f(x_n,y_n),\] \[k_2=h f\left(x_n+\frac{h}{2},y_n+\frac{k_1}{2}\right),\] \[k_3=h f\left(x_n+\frac{h}{2},y_n+\frac{k_2}{2}\right),\] \[k_4=h f(x_n+h,y_n+k_3).\]Then
\[y_{n+1}=y_n+\frac{1}{6}(k_1+2k_2+2k_3+k_4).\]Why it works well
The method samples the slope at the beginning, middle, and end of the interval. The weighted average gives a good estimate of the net change in $y$.
One RK4 step
Use one fourth-order Runge-Kutta step for
\[\frac{dy}{dx}=x+y,\qquad y(0)=1,\]with $h=0.1$. Here $f(x,y)=x+y$.
\[k_1=0.1\,f(0,1)=0.1.\] \[k_2=0.1\,f(0.05,1.05)=0.1(1.10)=0.110.\] \[k_3=0.1\,f(0.05,1.055)=0.1(1.105)=0.1105.\] \[k_4=0.1\,f(0.1,1.1105)=0.1(1.2105)=0.12105.\]Therefore
\[y(0.1)\approx 1+\frac{1}{6} (0.1+2(0.110)+2(0.1105)+0.12105).\]Hence
\[\boxed{y(0.1)\approx 1.11034.}\]Key points
- Runge-Kutta methods are step-by-step methods.
- The fourth-order method is accurate for many smooth problems.
- Step size must be chosen carefully: too large gives error, too small may waste computation.
Practice questions
- Write the standard fourth-order Runge-Kutta formula.
- What is the role of $k_1$, $k_2$, $k_3$, and $k_4$?
- Use one RK4 step to solve $dy/dx=x+y$, $y(0)=1$, with $h=0.1$.
- Why is RK4 usually more accurate than Euler’s method?
- What happens if the step size is chosen too large?
Discussion