Explore BrainMass
Share

Explore BrainMass

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

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

    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 October 10, 2019, 6:31 am ad1c9bdddf
    https://brainmass.com/math/partial-differential-equations/first-order-differential-equations-partial-547247

    Attachments

    Solution Preview

    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 ...

    Solution Summary

    The first order differential equations, partial DE's and linear functions are provided. The expert uses separation of variables to solve the verhulst population problems.

    $2.19