### 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) = c

*y^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.