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.
msc
semester-i
numerical-methods
differential-equations
runge-kutta
Many physical systems are described by differential equations. 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.
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 $h$.
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 $y$.
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.
Discussion