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Math Help - change of variables

  1. #1
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    change of variables

    part (a)
    use the substitution u = (y-x) to find the general family of solutions of the differential equation:

    dy/dx = sin(y-x)

    part (b)
    show that if a new independant variable is defined by y = z^(1-n) then the differential equation:

    dz/dx + p(x)z = q(x)z^n where (n not equal to 1)

    becomes a linear differential equation in y(x)

    part (c)
    use part (b) to solve the initial value problem

    xz dz/dx = x^2 + 3z^2 where z(1) = 1
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  2. #2
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    Quote Originally Posted by razorfever View Post
    part (a)
    use the substitution u = (y-x) to find the general family of solutions of the differential equation:

    dy/dx = sin(y-x)

    [snip]
    I only have time to give you a start for part (a):

    \frac{du}{dx} = \frac{dy}{dx} - 1 \Rightarrow \frac{dy}{dx} = \frac{du}{dx} + 1. So the DE is:

    \frac{du}{dx} + 1 = \sin u.

    Therefore x = \int \frac{1}{\sin (u) - 1} \, du.

    Solve for x in terms of u and then substitute back u = y - x.
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