what is the general solution of a differential equation of the form dy/dx = C - Ky^4?

where C and K are constants

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  • hfshaw
    Lv 7
    1 decade ago
    Favorite Answer

    This is a separable equation:

    dy/dx = c - k*y^4 = -k*(y^4 - c/k)

    dy/(y^4 - c/k) = - k dx

    For clarity in what follows, let a^4 = c/k.

    Then:

    dy/(y^4 - a^4) = -k dx

    Expand the left hand side in terms of partial fractions:

    (1/(4a^3))*[1/(y-a) - 1/(y+a) - 2a/(y^2 + a^2) = -k dx

    Integrate:

    (1/(4a^3))*[ln(y-a) - ln(y+a) - 2*arctan(y/a) = -k*x + c

    where c is the constant of integration.

    ln((y-a)/(y+a)) - 2*arctan(y/a) = 4*(a^3)*(c - k*x)

    Backsubstituting for a:

    ln((y - (c/k)^(1/4))/(y + (c/k)^(1/4))) - 2*arctan(y/(c/k)^(1/4)) = 4*((c/k)^(3/4))*((c/k)^(1/4) - k*x)

    This is an implicit solution for y, and I I think this is as good as you are going to get. There does not appear to be a closed-form solution for y in terms of elementary functions.

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