Find the general solution of the given differential equation. ydx − 6(x + y⁸) dy = 0

Solution:

Given ydx − 6(x + y⁸)dy = 0

ydx = 6(x + y⁸)dy

dx ={ [6x + 6y⁸ ]/y }.dy

dx/dy = 6x/y + 6y⁷

dx/dy - 6x/y = 6y⁷

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 = 6y⁷

Integrating factor = I.F. = e^∫pdy = e^∫6/ydy = e^-6logy  = 1/y⁶

Solution is (I.F.) x = ∫ I.F. q. dy

(1/y⁶)x = ∫ (1/y⁶) 6y⁷dy

x/y⁶ = 6y²/2 + C

x/y⁶ = 3y² + C

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Find the general solution of the given differential equation. ydx − 6(x + y⁸)dy = 0

Summary:

The general solution of the given differential equation ydx − 6(x + y⁸)dy = 0 is x/y⁶ = 3y² + C.