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Find all solutions of the differential equation y′′ – 2y′ + 5y = 0

Find all solutions of the second-order linear differential equation

    \[ y'' - 2y' + 5y = 0 \]

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


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

    \[ y'' + ay' + by = 0 \qquad \text{with} \qquad a = -2, \quad b = 5.\]

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

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

Therefore,

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

for constants c_1 and c_2.

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