Find the general solution of y'' + 4y = 0

Find the general solution of $y'' + 4y = 0$.

Step 1: Characteristic equation

Substituting $y=e^{rx}$: $r^2+4=0.$

There is no real $r$-term, so the roots are purely imaginary: \[r^2=-4 \implies r=\pm2i \implies \alpha=0,\ \ \beta=2.\]

Step 2: Fundamental system

Purely imaginary roots $r=\pm i\beta$ give the real fundamental system $\cos\beta x,\ \sin\beta x$: $y_1=\cos 2x, \qquad y_2=\sin 2x.$

Step 3: Wronskian

\[ W(y_1,y_2)=\begin{vmatrix}\cos 2x & \sin 2x\\ -2\sin 2x & 2\cos 2x\end{vmatrix} = 2\cos^2 2x+2\sin^2 2x = 2 \neq 0, \]

confirming $y_1,y_2$ are linearly independent for all $x$.

Step 4: General solution

The solution is recorded in the result below.