## MIT 1803 lecture 12

This lecture covered introduced the second order inhomogeneous equation .

Theorem: where is a linear operator.

Solution: that is a particular solution plus a characteristic solution. The method for solving is: first find , and then find . is a particular solution to .

Prove: All the are solutions, by,

From above we know that so the remainder is .

Prove: There are no other solutions, by, let be a solution, then

He is so careful in even choosing the names of constants – they should suggest a system or be intentionally neutral, suggesting nothing.

Lastly, he looks at the question – can second order linear inhomogeneous equations have transient solutions? That is, under what conditions of and does go to zero as ? He lists the characteristic roots, solutions, and stability conditions, and concludes that the simplist most elegant way to say it is **the ODE is stable if all characteristic roots have a negative real part**.

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