# Find the general solution of the given differential equation. ydx − 6(x + y^{8}) dy = 0

**Solution:**

Given ydx − 6(x + y^{8})dy = 0

ydx = 6(x + y^{8})dy

dx ={ [6x + 6y^{8 }]/y }.dy

dx/dy = 6x/y + 6y^{7}

dx/dy - 6x/y = 6y^{7}

It is a linear differential equation of the type dx/dy +p =q

where, p and q are functions of x

∴p = 6/y and q = 6y^{7}

Integrating factor = I.F. = e^{∫pdy} = e^{∫6/ydy} = e^{-6logy } = 1/y^{6}

Solution is (I.F.) x = ∫ I.F. q. dy

(1/y^{6})x = ∫ (1/y^{6}) 6y^{7}dy

x/y^{6} = 6y^{2}/2 + C

x/y^{6} = 3y^{2} + C

## Find the general solution of the given differential equation. ydx − 6(x + y^{8})dy = 0

**Summary:**

The general solution of the given differential equation ydx − 6(x + y^{8})dy = 0 is x/y^{6} = 3y^{2} + C.

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