\section{Linear Second Order ODE/y'' + 4 y = 0}
Tags: Second Order ODE: y'' + 4 y = 0, Linear Second Order ODE: y'' + 4 y = 0, Examples of Second Order ODE, Examples of Constant Coefficient Homogeneous LSOODEs, Examples of Second Order ODEs

\begin{theorem}
The second order ODE:
:$(1): \quad y'' + 4 y = 0$
has the general solution:
:$y = C_1 \cos 2 x + C_2 \sin 2 x$
\end{theorem}

\begin{proof}
It can be seen that $(1)$ is a constant coefficient homogeneous linear second order ODE.
Its auxiliary equation is:
:$(2): \quad: m^2 + 4 = 0$
From Solution to Quadratic Equation: Real Coefficients, the roots of $(2)$ are:
:$m_1 = 2 i$
:$m_2 = -2 i$
These are complex and unequal.
So from Solution of Constant Coefficient Homogeneous LSOODE, the general solution of $(1)$ is:
:$y = C_1 \cos 2 x + C_2 \sin 2 x$
{{qed}}
Category:Examples of Second Order ODE
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\end{proof}

