Find all solutions of the differential equation y′′ + 4y′ = 0

by RoRi

January 31, 2016

Find all solutions of the second-order linear differential equation

$y'' + 4y' = 0$

on the interval $(-\infty, +\infty)$ .

The given second-order linear differential equation is of the form

$y'' + ay' + by = 0 \qquad \text{with} \qquad a = 4, \quad b = 0.$

These values of $a$ and $b$ give us $d = a^2 - 4b = 16$ . Hence, $d > 0$ and $k = \frac{1}{2}\sqrt{d} = 2$ . By Theorem 8.7 (page 326-327 of Apostol) we then have

$y = e^{-\frac{ax}{2}} \left( c_1 u_1(x) + c_2 u_2(x) \right) \qquad \text{where} \qquad u_1 (x) = e^{kx}, \quad u_2 (x) = e^{-kx}.$

Therefore,

$\begin{align*} y &= e^{-2x} \left( c_1 e^{2x} + c_2 e^{-2x} \right) \\ &= c_1 + c_2 e^{-4x} \end{align*}$

for constants $c_1$ and $c_2$ .

Stumbling Robot

Find all solutions of the differential equation y′′ + 4y′ = 0

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