Question

solve the differential equation dy/dx(x/1-x^2)y=xy^1/2 , y(0)=1

Answer 1

$y'\cdot \frac{x}{1-x^2}=xy^\frac{1}{2}, \quad y(0)=1\\ \frac{dy}{y^\frac{1}{2}}=(1-x^2)dx\\ \int\frac{dy}{y^\frac{1}{2}}=\int(1-x^2)dx\\ 2y^\frac{1}{2}=x-\frac{x^3}{3}+c\\ y(0)=1\implies y=1, x=0.$

Then

$2=0-0+c\implies c=2$

Solution of differential equation is

$2y^\frac{1}{2}=x-\frac{x^3}{3}+2$