Linear Second Order ODE/y'' - 2 y' + 5 y = 0
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Theorem
The second order ODE:
- $(1): \quad y - 2 y' + 5 y = 0$
has the general solution:
- $y = e^x \paren {C_1 \cos 2 x + C_2 \sin 2 x}$
Proof
It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
- $(2): \quad: m^2 - 2 m + 5 = 0$
From Solution to Quadratic Equation with Real Coefficients, the roots of $(2)$ are:
- $m_1 = 1 + 2 i$
- $m_2 = 1 - 2 i$
So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
- $y = e^x \paren {C_1 \cos 2 x + C_2 \sin 2 x}$
$\blacksquare$