Second-Order Differential Equations

Question 1

What is a second-order linear ODE in standard form?

Accepted Answer

An equation of the form $y''+p(x)y'+q(x)y=g(x)$, where $p,q,g$ are given functions of $x$ and $y$ is the unknown function.

Question 2

What is the characteristic equation for $ay''+by'+cy=0$ with constant coefficients?

Accepted Answer

Assuming $y=e^{rx}$ gives the characteristic equation $$ar^2+br+c=0,$$ where $r$ is the exponential growth/decay rate.

Question 3

State the variation of parameters formulas for a particular solution of $y''+p(x)y'+q(x)y=g(x)$ given $y_1,y_2$.

Accepted Answer

With fundamental solutions $y_1,y_2$ and Wronskian $W=y_1y_2'-y_2y_1'$, set $y_p=u_1y_1+u_2y_2$ where
$$u_1'=-\frac{y_2\,g(x)}{W},\qquad u_2'=\frac{y_1\,g(x)}{W}.$$

Second-Order Differential Equations

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