## MIT 1803 lecture 11

This lecture covered a **study of linearity** in y”, y’, y, etc. on the second order homogeneous differential equations. He addressed the questions

- Why are solutions of a second order linear ODE?
- What are they
*all*the solutions?

By answering carefully answering these questions **elegantly**, the theory can be developed to also handle higher order ODEs without any extra work.

To insure the work is elegantly extensible we first prove the **superposition principle** using operator notation. Let be the *differentiation operator* that operates on or applies to , and let be the *linear operator*. The definition of a linear operator is an operator that obeys the rules

(where is a constant). Now we can observe that is linear because

Notes that the second order ODE becomes

Imagine as a black box that takes in a function and outputs a function . Finding the solution to the homogeneous linear ODE is equivalent to asking “if we want , so what must we put into ?”

Proof of 2) starts by letting and giving initial conditions and , then

this set of (two) linear equations is solvable **iff** the **Wronskian** is

note that if (the solutions are not linearly independent) then (but note that the reverse is not strictly true – the Wronskian can be zero for other reasons).

Now, somehow, we are meant to find and which are **normalized solutions**, which are better than other solutions because their initial values are nicer. Why are normalized solutions so good? Because they allow us to instantly solve the initial conditions problem. This can be seen from

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