y = C1 cos (2x) + C2 sin (2x)
where C1 and C2 are arbitrary constants.
This can be found using the following steps:
Find the characteristic equation of the differential equation. The characteristic equation is found by replacing y with e
mx
in the differential equation. This gives us the equation:
m^2 + 4 = 0
Solve the characteristic equation for m. This gives us the following roots:
m = -2, 2
Since the roots of the characteristic equation are complex, the general solution of the differential equation is of the form:
y = C1 e^{-2x} + C2 e^{2x}
We can use Euler's formula to convert the exponential terms into trigonometric terms. This gives us the following general solution:
y = C1 cos (2x) + C2 sin (2x)
where C1 and C2 are arbitrary constants.
I hope this helps! Let me know if you have any other questions.