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# Differential equations

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Hi,

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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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 is

22. The auxiliary equation is

The roots are:

With complex roots, the general solution is

And the first derivative is

Now find c1 and c2 for the initial conditions y(0) = 1 and y'(0) = -1

So the partial solution satisfies the initial conditions is

34. substituting all parameters to the equation in (20) gives
(1)
Differential both sides gives

Since , we have a second order differential equation for I:

The auxiliary equation is

The roots are:

With complex roots, the general solution is

Since the initial current is 0, then

The current is then
(2)
And
(3)
Substituting (2) and (3) to (1) gives

When t = 0, the charge q(0) = 0, so substituting t = 0 and q(0) to above equation to solve for c2

Therefore,

38.

Thus

When t = 0

(b) the auxiliary equation is

Then the general solution to the equation is

Now find the first derivative

When t = 0, using the initial conditions

Thus, the particular solution with the given initial conditions is

(c) by the uniqueness, the solution to is

Substituting f(t) = sin(x + t) gives

Which is the formula for the sum of angles.

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