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We know from part c) that the general form of the solution is
with arbitrary constants
We first turn the solution into a real form, and then use the given initial condition to replace the arbitrary constants with specific numbers.
To turn the solution into real form, we replace the complex terms and appearing in the general solution with and .
Using the Euler relation we have:
So the complex term can be written as
Thus the real form of the general solution is:
for arbitrary real constants and .
Now we use the given initial data to solve for and . Remember that and
Putting the initial data at time into the general solution gives:
This gives the two equations
and which has solution
So the solution satisfying the initial data is