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

$y'' + by = 0 \qquad b = 4.$

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

$y = c_1 u_1(x) + c_2u_2(x), \quad \text{where} \quad u_1(x) = \cos (kx), \quad u_2(x) = \sin(kx),$

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

$y = c_1 \cos (2x) + c_2 \sin (2x)$

for some constants $c_1$ and $c_2$ .

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