Numerical Analysis/ODE in vector form Exercises

All of the standard methods for solving ordinary differential equations are intended for first order equations. When you need to solve a higher order differential equation, you first convert it to a system of first order of equations. Then you rewrite as a vector form and solve this ODE using a standard method. On this page we demonstrate how to convert to a system of equations and then apply standard methods in vector form.

Reduction to a first order systemEdit

(Based on Reduction of Order and Converting a general higher order equation.)

I want to show how to convert higher order differential equation to a system of the first order differential equation. Any differential equation of order n of the form


can be written as a system of n first-order differential equations by defining a new family of unknown functions


The n-dimensional system of first-order coupled differential equations is then


Differentiating both sides yields


We can express this more compactly in vector form


where   for   and   =  


Consider the second order differential equation   with initial conditions   and  . We will use two steps with step size   and approximate the values of   and  

Since the exact solution is   we have   and  .

Exercise 1: Convert this second order differential equation to a system of first order equations.Edit

Exercise 2: Apply the Euler method twice.Edit

Exercise 3: Apply the Backward Euler method twice.Edit

Exercise 4: Apply the Midpoint method twice.Edit

Exercise 5: Using the values from the Midpoint method at t = h in exercise3, apply the Two-step Adams-Bashforth method once.Edit