MIT 1803 lecture 12

This lecture covered introduced the second order inhomogeneous equation y'' +py' + qy = f(x) .

Theorem:  Ly = f(x) where L is a linear operator.

Solution: y_p + y_c that is a particular solution plus a characteristic solution.  The method for solving is:  first find y_c , and then find y_p y_p is a particular solution to Ly=f(x) .

y = y_p + c_1y_1 + c_2y_2

Prove:  All the y_p + c_1y_1 + c_2y_2 are solutions, by,

L(y_p + c_1y_1 + c_2y_2) = L(y_p) + L(c_1y_1 + c_2y_2)

From above we know that L(y_p)=f(x) so the remainder is L(c_1y_1 + c_2y_2) =0 .

Prove:  There are no other solutions, by, let u(x) be a solution, then

L(u) = f(x) \quad L(y_p) = f(x) \quad L(u-y_p) = 0

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 A and B does c_1y_1 + c_2y_2 go to zero as t \rightarrow \infty ?  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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