Differential Equation Solver
Solve first-order ODEs numerically using RK4 given dy/dx = f(x,y) and an initial condition.
Inputs
Enter the right-hand side using x and y.
y(target x)
3.43656366
Steps used
100
Step size h
0.01000000
Step by step
Values used
dy/dx = f(x, y) = x + y; Initial x₀ = 0; Initial y₀ = y(x₀) = 1; Target x = 1; Number of steps = 100
RK4 method
y(n+1) = y(n) + (h/6)(k1 + 2k2 + 2k3 + k4)
y(target x)
= 3.43656366
Steps used
= 100
Step size h
= 0.01000000
How it works
Solves the initial value problem dy/dx = f(x,y), y(x₀) = y₀ using the classical 4th-order Runge-Kutta method (RK4). This method has local error O(h⁵) and global error O(h⁴), making it highly accurate for smooth ODEs. Enter f(x,y) using x and y as variables.
Formula
RK4 method
y(n+1) = y(n) + (h/6)(k1 + 2k2 + 2k3 + k4)
- k1
- h·f(xn, yn)
- k2
- h·f(xn+h/2, yn+k1/2)
- k3
- h·f(xn+h/2, yn+k2/2)
- k4
- h·f(xn+h, yn+k3)
Frequently Asked Questions
How many steps should I use?
More steps give higher accuracy but take longer. For most problems, 100-1000 steps suffice. If the solution diverges, try more steps (smaller h).
Can this solve higher-order ODEs?
Convert a second-order ODE y'' = g(x,y,y') into a system of first-order ODEs by substituting v = y', then solve iteratively.