Numerical Analysis/ODE in vector form Exercises

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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 system[edit]

(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 =

Exercise[edit]

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 in exercise3, apply the Two-step Adams-Bashforth method once.[edit]

Reference[edit]

http://en.wikipedia.org/wiki/Ordinary_differential_equation

http://www.math.ohiou.edu/courses/math3600/lecture29.pdf

http://www.ohio.edu/people/mohlenka/20131/4600-5600/hw7.pdf