Help me with Differential Equations?

Question

Solve the differential equation y"+4y=0.

r^2+4=0  

r=2i, -2i

Now what?

Answer 1

When you get complex roots a ± bi, the general solution becomes y(t) = e^atcos(bt) + e^atsin(bt). So for this problem you get y(t) = cos(2t) + sin(2t)

Answer 2

You have applied Euler's method to solve the homogeneous with constant coefficeints.

 The common solution of the equation can be expressed like this:

                                               y = C₁ e ^rx + C₂ e ^-rx                                                 (1)

You have found the right equation for r after substituting  (1) into you differential equation. Use

                                                           r = 2i and r = -2i 

to come up with final solution:

                                                          y = C₁ e ^2ri  + C₂ e ^{- 2ri}

This solution can be expressed in terms of trigonometric functions, but coefficients C₁ and C₂ have to be found from boundary conditions which are not given here.