integrate-1-x-2-y-xy-2x-0-with-cond-y-1-0-

Question Number 30419 by abdo imad last updated on 22/Feb/18

i n t e g r a t e (1 + x^{2}) y^{^{'}} + x y - 2 x = 0 w i t h c o n d . y (1) = 0

Answered by sma3l2996 last updated on 24/Feb/18

(1 + x^{2}) y^{'} = x (2 - y)

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

- l n ∣ 2 - y ∣= \frac{1}{2} l n ∣ C (1 + x^{2}) ∣

\frac{1}{2 - y} = \sqrt{C (1 + x^{2})}

y = 2 - \frac{1}{\sqrt{C (1 + x^{2})}}

y (1) = 0 \Rightarrow \frac{1}{\sqrt{2 C}} = 2 \Leftrightarrow C = \frac{1}{8}

s o y (x) = 2 (1 - \frac{\sqrt{2}}{\sqrt{1 + x^{2}}})