Principle of Superposition

From Problem Set 5: Exercise 1.2

To explore this Principle and its implications, we must first set up the environment where it is used, thus let us define it.

Let us consider a linear operator embodying the for of the ODE in

Let there be two solutions to this form (where ) in the solutions y1 and y2.

The principle of superposition states that the linear combination of the two solutions of the two is in the form of . Here, c1 and c2 are arbitrary constants.

This solution is also in applicable to ODE for any values of c1 and c2, saying that it is a fundamental set.

We can show this by

This gives us

Taking constants common and constructing an equation

L can be substituted here from the above.

Thus we can see that the solutions are a linear combination of the two solutions with these scalar multiples of c1 and c2.

If c1 and c2 are not chosen carefully, there may be other solutions.