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    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.

    © 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

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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.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © 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

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