# Differential equations

Hi,

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.

https://brainmass.com/math/calculus-and-analysis/differential-equations-examples-initial-value-problem-208591

## SOLUTION This solution is **FREE** courtesy of BrainMass!

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 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.

Â© BrainMass Inc. brainmass.com September 28, 2022, 10:30 pm ad1c9bdddf>https://brainmass.com/math/calculus-and-analysis/differential-equations-examples-initial-value-problem-208591