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Help me with Differential Equations?

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2 Answers

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

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.