Explore BrainMass

Explore BrainMass

    solving differential equations

    Not what you're looking for? Search our solutions OR ask your own Custom question.

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

    Please see the attached file for full description.

    1. Find both the first and second order differentials (y' and y") for the following functions:

    2. Use integrating factor to convert the following equation into "exact ODE" form and solve for y.
    2xy' = (y -x) (y + x)/ y

    3. Solve the differential equation, y is a function of x
    y'' - 6y' + 9y = x^2x

    4. Solve the differential equation, y is a function of x
    2y'' - 2y' + 5y = cos(x), y(0) = 1, y'(0) = 2

    5. Solve the differential equation, y is a function of x
    y'= 2x + 2xy, y(1)= 2

    © BrainMass Inc. brainmass.com December 24, 2021, 10:00 pm ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/solving-differential-equations-433751

    Attachments

    SOLUTION This solution is FREE courtesy of BrainMass!

    ET 7430

    1. Find both the first and second order differentials (y' and y") for the following functions:
    
    a.
    Using the product rule and chain rule:

    Then the second derivative is:

    b.

    Then the second derivative is:

    c.
    Apply the chain rule:

    Apply the product rule and chain rule:

    2. Use integrating factor to convert the following equation into "exact ODE" form and solve for y.

    Rearrange the equation:

    
    Then

    And
    is a function of x alone, then
    is an integrating factor.
    So multiplying 1/x2 to the equation (1):

    (2)
    Whose solution is obtained as follows:

    Then

    Where is the coefficient of dx in equation (2)
    So

    Therefore,

    And

    Which implies that

    And the general solution is

    3. Solve the differential equation, y is a function of x

    The characteristic equation of the homogeneous ODE is

    It has double roots
    Then the general solution to the homogeneous ODE is

    Now, since the right side function is an exponential and we know that exponentials never just appear or disappear in the differentiation process it seems that a likely form of the particular solution would be

    Now, all that we need to do is do a couple of derivatives, plug this into the differential equation and see if we can determine what A needs to be.

    Then

    So a particular solution to the differential equation is then,

    Thus, the general solution to the differential equation is

    4. Solve the differential equation, y is a function of x

    For this undetermined coefficient type of ODE, since the right-hand function is , we guess that the particular solution is

    Now differentiate it:

    Substituting back gives:

    Compare the coefficients, we have

    Solve for A and B:

    So the particular solution is

    Now find the general solution to the homogenous equation
    The roots of its characteristic equation are:

    Then general solution to the homogenous ODE is

    Therefore, the general solution to the given ODE is:

    Now find the constants c1 and c2 using the initial values:

    Differentiate y(x):

    Then

    Thus the solution of this IVP is:

    5. Solve the differential equation, y is a function of x

    So this DE can be solved by separation of variables.
    Integrate on both sides:

    Then

    Substitute 1 for x to find constant A:

    So the solution is

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

    © BrainMass Inc. brainmass.com December 24, 2021, 10:00 pm ad1c9bdddf>
    https://brainmass.com/math/calculus-and-analysis/solving-differential-equations-433751

    Attachments

    ADVERTISEMENT