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

$y'' + by = 0 \qquad \text{with} \qquad b = -4.$

By Theorem 8.6 (page 326 of Apostol) we then have

$y = c_1 u_1(x) + c_2u_2(x), \quad \text{where} \quad u_1(x) = e^{kx}, \quad u_2(x) = e^{-kx},$

with $b = -4$ implies $k = 2$ . Hence, all solutions are given by

$y = c_1 e^{2x} + c_2 e^{-2x}$

for some constants $c_1$ and $c_2$ .

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