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    First Order Differential Equations, Partial DE's/ Linear Dependence/Independance

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    1. Find solutions to the given Cauchy- Euler equation
    (a) xy'+ y =0 (b) x2y'' + xy'+y =0 ; y(1) =1, y'(1) =0

    2. Find a solution to the initial value problem
    x2y' + 2xy = 0; y (1) = 2

    3. Find the general solution to the given problems
    (a) Y' + (cot x)y = 2cosx (b) (x-5)(xy'+3y) = 2

    4. Solve the Bernoulli equation y' = xy3 - 4y

    5. Use separation of variables to solve the verhulst population problem

    N' (t) = (a-bN) N, N (0) = N0; a,b > 0

    6. Verify that each of the given functions is a solution of the given differential equation, and then use the Wronskian to determine linear dependence/ independence
    Y''' - y''- 2y' = 0 {1, e-x, e2x}

    © BrainMass Inc. brainmass.com December 24, 2021, 11:12 pm ad1c9bdddf
    https://brainmass.com/math/partial-differential-equations/first-order-differential-equations-partial-547247

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    SOLUTION This solution is FREE courtesy of BrainMass!

    1a.
    The equation is:
    (1.1)
    We "guess" a solution in the form:
    (1.2)
    We substitute this back in the equation:

    (1.3)
    Thus the solution to (1.1) is:
    (1.4)
    Where C is a constant to be determined from initial conditions.

    1b.
    The equation is:
    (1.5)
    We "guess" a solution in the form:
    (1.6)
    We substitute this back in the equation:

    (1.7)
    We have a single pure complex root with multiplicity of two Thus the solution to (1.5) is a linear combination
    (1.8)
    From initial conditions:


    Thus:
    (1.9)

    2.
    The equation is:
    (1.10)
    Note that we can write the left hand side as:
    (1.11)
    Thus, the equation is:
    (1.12)
    Integrating both sides:
    (1.13)
    Where C is a constant.
    The general solution is:
    (1.14)
    Applying initial conditions:
    (1.15)
    The solution is:
    (1.16)

    3a.
    The equation is:
    (1.17)
    We rewrite the equation by multiplying both sides by sin(x):
    (1.18)
    And we note that the left hand side can be written as:
    (1.19)
    And the right hand side is:
    (1.20)
    Therefore:
    (1.21)
    Integrating both sides:

    (1.22)

    3b
    The equation is:
    (1.23)
    We rewrite the equation:
    (1.24)
    The integration factor is:
    (1.25)
    We multiply both sides by and we get:
    (1.26)
    The left hand side is now:
    (1.27)
    We integrate both sides:

    (1.28)
    Where c is an integration constant.

    We use partial fraction conversion:

    (1.29)
    Equating the coefficients of the numerator:

    (1.30)
    Therefore:

    (1.31)

    4.
    The equation is:
    (1.32)
    Rewriting:
    (1.33)
    This is a Bernoulli with n=3, so we set:
    (1.34)
    Thus:
    (1.35)
    Hence:

    (1.36)
    This is a simple first order equation.
    The integration factor is:
    (1.37)
    We multiply both sides by the integration factor:

    (1.38)
    Integration of both sides yields:
    (1.39)
    Integrating by parts:
    (1.40)
    Thus:

    (1.41)
    So now we can use (1.34):


    So the solution to (1.32) is:
    (1.42)

    5.
    The equation is:
    (1.43)
    Separating the equation:
    (1.44)
    Now each side is a function of a single variable, so we can integrate.
    We use partial fractions:

    Equating the numerators:
    (1.45)
    Therefore:
    (1.46)

    So when we integrate both sides of (1.44) we get:

    (1.47)

    Applying initial conditions:

    (1.48)
    Thus

    (1.49)

    6.
    The equation is:
    (1.50)
    Solution 1:
    Then indeed it is a solution since
    (1.51)
    Solution 2:
    Then:
    (1.52)
    Plugging it back into the equation yields:

    So it is a solution.


    Solution 2:
    Then:
    (1.53)

    Plugging it back into the equation yields:

    So is a solution.
    The Wronskian is:
    (1.54)
    Since there is no where , all the solutions are independent.

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

    © BrainMass Inc. brainmass.com December 24, 2021, 11:12 pm ad1c9bdddf>
    https://brainmass.com/math/partial-differential-equations/first-order-differential-equations-partial-547247

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