How to solve this differential equation

d^2y /dx^2 + 4y = 0

d²y/dx² + 4y = 0

is a homogeneous second order linear differential equation.

It's characteristic equation is:

r² + 4 = 0

r² = -4

r = ±2i

This has a general solution of y = e^(ax){(C1)cos(bx) + (C2)sin(bx)} where the roots are a ± bi

Since a = 0 the solution is:

y = (C1)cos(2x) + (C2)sin(2x)

You need boundary values to solve for the constants C1 and C2

You are welcome :D

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Its fine douglas! your explanation is fine

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This is a homogeneous linear ordinary differential equation of the second order.

Characteristic polynomial: r^2 + 4
Characteristic equation: r^2 + 4 = 0
Characteristic roots: r = -2j

Apply Euler's formula for imaginary roots, which yields the general solution:

y(t) = c1 * cos(2t) + c2 * sin(2t)

the aux equation is

r^2 + 4 = 0
r = ( + or - ) 2i
y = (c1) sin2x + (c2) cos2x