Reduction Of Second Order Equation To A First Order System

Reduction of second order equation to a first order system :

Some second-order equations can be reduced to first-order equations, rendering them susceptible to the simple methods of solving equations of the first order.

Let the second order initial value problem be given as

.....................1.1

We can reduce this second order initial value problem to a system of two first order equations.

Define u₁ = y. Then, we have the system

The system is given by

...................1.2

where,

In general, we may have a system as

.....................1.3

In vector notation, denote

Then, we can write the system as

..................1.4

Therefore, the methods derived for the solution of the first order initial value problem

.................1.5

can be used to solve the system of equations (1.4) or (1.5), that is, the second order initial value problem (1.1), by writing the method in vector form