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

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(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 =


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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.

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Exercise 2: Apply the Euler method twice.

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Exercise 3: Apply the Backward Euler method twice.

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Exercise 4: Apply the Midpoint method twice.

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

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