11-17-25 Second-order linear differential equations

A second-order linear differential equation is a differential equation of the form:

• $a(x)y^{\prime\prime} + b(x)y^\prime + c(x)y = g(x)$

• $a(x) \ne 0$

If the right-hand-side is 0, then the equation is “homogenous”.

Example. Solve the following equation

• $y^{\prime\prime} - 3y^\prime + 2y =0$

with initial conditions $y(0) = 1$, $y^\prime(0) = 0$

$r²-3r+2=0 \Rightarrow (r-1)(r-2)=0$

$r =1$ and $r=2$

• $y(t) = C_1e^y+C_2e^{2t}$