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Runge-Kutta Method for Approximate Solution of First Order Differential Equations

In a Nut Shell: The Runge-Kutta Method is a "fourth order" method and provides

greater accuracy than Picard's or Euler's Methods although it involves more extensive

calculations.

Problem Statement: Find an approximate solution to the first order, ordinary d.e.

dy/dx = f [ (x,y(x) ] subject to the condition y(0) = y_o in [a,b]

Strategy: Use the Runge-Kutta Algorithm as given below

y_n+1= y_n + h k n ≥ 0

where h = step size

k = (1/6) [ k₁ + 2k₂ + 2k₃ + k₄ ]

and k₁ = f(x_n, y_n) , k₂ = f( x_n + h/2 , y_n + hk₁ / 2 )

k₃ = f( x_n + h/2 , y_n + hk₂ / 2 ) , k₄ = f( x_n+1 , y_n + hk₃ )

Example: Use the Runge-Kutta Method to find an approximate solution to

dy/dx = 2 y , y(0) = 1/2 at x = 0.5 using step size, h = 0.25

The exact solution is y(x) = (1/2) e^2x .

Compare the approximate solution with the exact solution at x = 0.5.

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