Second Order Differential Equations

The following are general solutions of second order linear differential equations with constant coefficients

Let the differential equation be

\( a \frac{d^2y}{dx^2}+b \frac{dy}{dx}+c= 0 \)

Then the auxiliary equation is

\( a \lambda^2+b \lambda+c=0\)

Consider the auxiliary equation. If:

(Case 1)

\begin{align} b^2-4ac \gt 0\,,\ with\ distinct\ roots\ \alpha\ and\ \beta \\ then\ the\ solution\ to\ the\ differential\ equation\ is \\ y=Ae^{\alpha x} +Be^{\beta x} \end{align}

(Case 2)

\begin{align} b^2-4ac = 0\,,\ with\ repeated\ root\ \alpha \\ then\ the\ solution\ to\ the\ differential\ equation\ is \\ y=e^{\alpha x}(A+ Bx) \end{align}

(Case 3)

\begin{align} b^2-4ac \lt 0\,, with\ complex\ roots\ p\pm qi \\ the\ solution\ to\ the\ differential\ equation\ is \\ y=Ae^{px}cos(qx+\epsilon) \\ \\ And\ because \ Acos(qx+\epsilon)=Bcos(qx) +Csin(qx) \\ an\ alternative\ form\ of\ the\ solution\ is \\ y= e^{px}(Bcos(qx) +Csin(qx)) \end{align}