The given differential equation is:

(x^7y^2 - 3y)dx + (3x^8y - x)dy = 0


To solve this differential equation, we can use an integrating factor. The integrating factor is a function that multiplies both sides of the equation to make it exact.

The integrating factor (IF) can be found by multiplying the equation by an appropriate function, such that the resulting equation becomes exact. Let's assume the IF is of the form xmyn.

Multiplying the given equation by the integrating factor, we get:

xmyn(x^7y^2 - 3y)dx + xmyn(3x^8y - x)dy = 0


To check if the equation becomes exact after multiplying by the integrating factor, we need to check if the following condition is satisfied:

(dM/dy) = (dN/dx)

Here, M = xmyn(x^7y^2 - 3y) and N = xmyn(3x^8y - x).

Differentiating M with respect to y and N with respect to x:

(dM/dy) = xmyn(2x^7y - 3) + xmyn(x^7y^2 - 3y) * (d/dy)(xmyn)
= xmyn(2x^7y - 3) + xmyn(x^7y^2 - 3y) * (n/x)

(dN/dx) = xmyn(24x^7y - 1) + xmyn(3x^8y - x) * (d/dx)(xmyn)
= xmyn(24x^7y - 1) + xmyn(3x^8y - x) * (m/y)

Setting (dM/dy) = (dN/dx), we have:

xmyn(2x^7y - 3) + xmyn(x^7y^2 - 3y) * (n/x) = xmyn(24x^7y - 1) + xmyn(3x^8y - x) * (m/y)

Simplifying the equation, we get:

2x^8y - 3x^7 + x^8y^2 - 3xy = 24x^8y - xm/y + 3x^8y^2 - xy

(2 + x)y - 3 + (x + 3x^8)y^2 = 24x^8y - xm/y + xy

Comparing the coefficients of y^2, y, and the constant term on both sides, we get:

2 + x = -xm/y
x + 3x^8 = 0
-3 = 0


From the second equation, we have:

x + 3x^8 = 0
x(1 + 3x^7) = 0

This gives us two possible values for x: x = 0 or x = -1/3^(1/7).

Substituting x = 0 into the first equation, we get