Find all solutions of the differential equation y′′ – 2y′ + y = 0

by RoRi

January 31, 2016

Find all solutions of the second-order linear differential equation

$y'' - 2y' + y = 0$

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

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

$y'' + ay' + by = 0 \qquad \text{with} \qquad a = -2, \quad b = 1.$

These values of $a$ and $b$ give us $d = a^2 - 4b = 0$ . By Theorem 8.7 (page 326-327 of Apostol) we then have

$y = e^{-\frac{ax}{2}} \left( c_1 u_1(x) + c_2 u_2(x) \right) \qquad \text{where} \qquad u_1 (x) = 1, \quad u_2 (x) = x.$

Therefore,

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

for constants $c_1$ and $c_2$ .

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