differential equation using separable variable method of

Question

dy/y=dx/x^2(1-x^2)

Answer 1

dy/y=dx/x^2(1-x^2)

Using separable variable

∫dy/y=∫dx/x^2(1-x^2)

ln y=∫dx/x^2(1-x^2)

Calculate∫dx/x^2(1-x^2)

1/x^2(1-x^2)=A/x+B/x^2+C/1+x+D/1-x

                     =Ax(1-x^2)+B((1-x^2))+C(x^2*(1-x))+D(x^2*(1+x))/x^2(1-x^2)

1=Ax(1-x^2)+B((1-x^2))+C(x^2*(1-x))+D(x^2*(1+x))

1=AX-Ax^3+B-Bx^2+Cx^2-Cx^3+Dx^2+Dx^3

1=(-A-C+D)x^3+(-B+C+D)x^2+Ax+(B)

A=0,B=1,C=-1/2 andD=1/2

∫1/x^2(1-x^2)=∫1/x^2+∫1/2(1+x)+∫1/2(1/1-x)

                        =-1/x-(1/2)ln(1+x)+(1/2)ln(1-x)

ln y=-1/x-(1/2)ln(1+x)+(1/2)ln(1-x)+C