calculus
posted by Amy on .
I posted this below, and no one answered. Please help with this very complicated question! Using 3(x3)(x^26x+23)^1/2, as the chain rule differentiation of f(x)=(x^26x+23). Please explain how I find the general solution to the differential equation dy/dx= 2/27(x3)SQUARE ROOT BEGINS (x^26x+23)/y SQUARE ROOT ENDS (y>0), giving answer in implicit form. I need details! Many thanks.

Separate the variables:
27sqrt(y)dy=2(x3)sqrt(x^26x+23)dx
Let z=x^26x+23, then dz=(x^26x+23)'dx=
(2x6)dx=2(x3)dx
27sqrt(y)dy=sqrt(z)dz
Integrating both sides gives
18y*sqrt(y)=(2/3)*z*sqrt(z)+C
18y*sqrt(y)=(2/3)*(x^26x+23)*sqrt(x^26x+23)+C