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

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Dr. Juanita Wolff

Feb 20, 2025

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^6This 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^7This 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.... Show More
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