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Differential equations

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Please help solving these problems

section 4.3 # 4,16,22,34,38

See attached

The auxiliary equation for the given differential equation has complex roots. Find a general solution.
Solve the given initial value problem.
Prove the sum of angle formula for the sine function by following these steps.

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Solution Summary

This provides several examples of working with differential equations, including initial value problem, complex roots, a proof of the sum of angle formula, and an RLC series circuit.

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Please see the attached file for detailed solution.

4. if the auxiliary equation has complex roots , the general solution has the form of , where C1 and C2 are both constants
The auxiliary equation is

The roots are:

Then the general solution is

16. The auxiliary equation is

The roots are:

Since the auxiliary equation has two district real roots, the general solution ...

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