the given equation can be written as
1 y d y d x + 2 x y 2 ( 1 − x 2 ) = x \frac{1}{\sqrt{y}} \frac{dy}{dx} + \frac{2x\sqrt{y}}{2(1-x^2)} = x y 1 d x d y + 2 ( 1 − x 2 ) 2 x y = x .....(1)
this is of the form f'(y) y' + P f(y) = Q
where P = x 2 ( 1 − x 2 ) \frac{x}{2(1-x^2)} 2 ( 1 − x 2 ) x
Q= x
f(y) = 2 y \sqrt{y} y
let v= 2y \sqrt{y} y
differentiating,
1 y d y d x = d v d x \frac{1}{\sqrt{y}} \frac{dy}{dx} = \frac{dv}{dx} y 1 d x d y = d x d v
(1) becomes
d v d x + x v 2 ( 1 − x 2 ) = x \frac{dv}{dx} + \frac{xv}{2(1-x^2)} = x d x d v + 2 ( 1 − x 2 ) xv = x
this is of the form d v d x + A v = B \frac{dv}{dx} + Av = B d x d v + A v = B
A = x 2 ( 1 − x 2 ) a n d B = x A = \frac{x}{2(1-x^2)} and B=x A = 2 ( 1 − x 2 ) x an d B = x
integrating factor = e ∫ P d x = e ∫ x 2 ( 1 − x 2 ) d x = ( 1 − x 2 ) ( − 1 / 4 ) e^{\intop Pdx} = e^{\intop \frac{x}{2(1-x^2)}dx} = (1-x^2)^{(-1/4)} e ∫ P d x = e ∫ 2 ( 1 − x 2 ) x d x = ( 1 − x 2 ) ( − 1/4 )
the general solution is given by:
v (1-x2 )-1/4 = ∫ \int ∫ x. (1-x2 )-1/4 dx + c
2y \sqrt{y} y = -2/3 (1-x2 ) + c(1-x2 )1/4
now, given y(0)=1
2 = -2/3 + c
c= 8/3
so, the solution is
2y \sqrt{y} y = -2/3 (1-x2 ) + 8/3(1-x2 )1/4
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