Finding a Basis for solutions of Second Order ODE's

In the last post, said I would show you how solutions of second-order linear, differential equations behave physically. But before we can do that, I have to cover this rather dry, math intense proof, in order to make sense of one of the 3 possible solutions. We'll discuss these possibilities in the next post, so I apologise for that...

Now in the last post, we found that sin(x) and cos(x) are linearly independent functions that satisfy the second order differential equation...

And hence the general solution to equation (1) is a linear combination of both of these functions. So we say that sin(x) and cos(x) form the basis of solutions for equation (2) on the interval (-∞, ∞).

The general solution to (1) is...

Reduction of order and finding a Basis if 1 solution is known...

The standard form of second-order Ordinary Differential Equations (ODE's) is shown in equation (2) below...

Like above with equation (1), we expect to find 2 linearly independent solutions for equation (2) in order to form a basis for the general solution.

Now let's assume we know one solution and want to find the other linearly independent solution...

To do this, we let...

then...

Substituting these derivatives into equation (2), we get...

Ok. Since is a solution to (2), the term in the last brackets is 0. Thus...

Now divide equation (3) by ...

Therefore equation (3) has been reduced to a first order ODE for which we can separate the variables and solve by integrating both sides...

We don't need a constant of integration because we are just finding the other solution to the ODE. So exponentiating both sides...

Now, since and , it follows that...

Let's see if it works...

Since we already know the basis of equation (1), let's see if all of this derivation works.

For equation (1), with and ...

Finally...

Therefore, we have found the other solution of sin(x). So, it goes to show that if we know a solution to a second-order ODE, we can find the other solution using equation (4).

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Credits:

All equations in this tutorial were created with QuickLatex

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2. Differential Equations: Order and Linearity
3. First-Order Differential Equations with Separable Variables - Example 1
4. Separable Differential Equations - Example 2
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Second Order Differential Equations

1. Introduction to Second Order Differential Equations
2. Finding a Basis for solutions of Second Order ODE's

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