Question

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

Answer 1

Given question

y'{1−x2x​ } = xy^1/2

$\implies$ $dy\over \sqrt{y}$ =(1-x²)dx

$\implies$ $\intop$ $dy\over \sqrt{2}$ =$\intop$ (1-x²)dx

$\implies$ 2y^1/2 = x - $x^3\over 3$ + C

Given y(0) = 1

$\implies$ at x= 0 , y = 1

$\implies$ 2 = 0 - 0 + C

$\rightarrow$ C=2

Answer

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