Find the general solution of ydx − 5(x + y^7) dy = 0?

1 Answer

  • y dx − 5(x + y^7) dy = 0

    This is more easily solved for x(y), because this dx/dy is a linear ODE. We will get an explicit solution for x(y), which may or may not be able to be rearranged to form an explicit solution for y(x).

    dx/dy − 5x/y = -5y^6

    This can be solved using an integating factor, p(y)

    p(y) = exp(INTEGRAL of {-5/y dy})

    p(y) = exp(-5ln(y)) = exp(ln(1/y^5)) = 1/y^5

    The solution for x(y) is then given by:

    x(y) = (y^5)INTEGRAL of {(1/y^5)(-5y^6) dy}

    x(y) = (y^5)*INTEGRAL of {-5y dy}

    x(y) = (y^5)*[c – (5/2)y^2]

    where c is the constant of integration.

    x(y) = cy^5 – (5/2)y^7

    This is an explicit solution for x(y), and an implicit solution for y(x). You cannot express an explicit solution for y(x) in closed form in terms of elementary functions.

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