Can someone solve this :)?.... dx+(x-1/2x^3y)dy=0

1 Answer
May 11, 2018

#"The GS is "1/3x(2x^2+3x+6)+2ln(x-1)+lny=c#.

Explanation:

I hope, the Question is to find the Gen. Soln. (GS) of the Diff. Eqn.:

#dx+(x-1)/(2x^3y)dy=0#.

We rewrite it as, #(2x^3)/(x-1)dx+dy/y=0#, and immediately

identify it as separable variable type.

Its GS is obtained by integrating term-wise.

#:. int(2x^3)/(x-1)dx+intdy/y=c#.

#:. 2int{(x^3-1)+1}/(x-1)dx+lny=c#,

#:. 2int{((x-1)(x^2+x+1)+1)/(x-1)}dx+lny=c#,

#:. 2int{(x^2+x+1)+1/(x-1)}dx+lny=c#,

#:. 2{(x^3/3+x^2/2+x)+ln(x-1)}+lny=c#,

#:. 2/6(2x^3+3x^2+6x)+2ln(x-1)+lny=c#,

#:. 1/3x(2x^2+3x+6)+2ln(x-1)+lny=c#, is the desired

GS!