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.