## 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.