solve-1-x-2-y-xy-1-x-2-

Question Number 90963 by abdomathmax last updated on 27/Apr/20

s o l v e (1 + x^{2}) y^{^{'}} + x y = \sqrt{1 + x^{2}}

Commented by niroj last updated on 27/Apr/20

(1 + x^{2}) y^{^{'}} + xy = \sqrt{1 + x^{2}}

\frac{dy}{dx} + \frac{x}{1 + x^{2}} y = \frac{\sqrt{1 + x^{2}}}{1 + x^{2}}

\frac{dy}{dx} + \frac{x}{1 + x^{2}} y = \frac{1}{\sqrt{1 + x^{2}}}

form \frac{dy}{dx} + Py = Q

let, P = \frac{x}{1 + x^{2}}, Q = \frac{1}{\sqrt{1 + x^{2}}}

IF = e^{\int Pdx} = e^{\int \frac{x}{1 + x^{2}} dx}

Put, x^{2} + 1 = t

2 xdx = dt

xdx = \frac{dt}{2}

IF = e^{\int \frac{1}{t} . \frac{dt}{2}} = e^{\frac{1}{2} \log t} = e^{\log \sqrt{t}}

∴ IF = \sqrt{t} = \sqrt{1 + x^{2}}

Complete solution

y . IF = \int IF . Qdx + c

y \sqrt{1 + x^{2}} = \int \sqrt{1 + x^{2}} . \frac{1}{\sqrt{1 + x^{2}}} dx + C

y \sqrt{1 + x^{2}} = \int dx + C

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

Commented by mathmax by abdo last updated on 27/Apr/20

(h e) \to (1 + x^{2}) y^{^{'}} + x y = 0 \Rightarrow (1 + x^{2}) y^{^{'}} = - x y \Rightarrow \frac{y^{^{'}}}{y} = - \frac{x}{1 + x^{2}} \Rightarrow

l n ∣ y ∣ = - \frac{1}{2} l n (1 + x^{2}) + c \Rightarrow y (x) = \frac{K}{\sqrt{1 + x^{2}}} l e t u s e m v c m e t h o d

y^{^{'}} = \frac{K^{^{'}}}{\sqrt{1 + x^{2}}} + K (- \frac{1}{2} (2 x) {(1 + x^{2})}^{- \frac{3}{2}}) = \frac{K^{^{'}}}{\sqrt{1 + x^{2}}} - K x {(1 + x^{2})}^{- \frac{3}{2}}

(e) \Rightarrow \sqrt{1 + x^{2}} K^{^{'}} - \frac{K x}{\sqrt{1 + x^{2}}} + \frac{x K}{\sqrt{1 + x^{2}}} = \sqrt{1 + x^{2}} \Rightarrow

K^{^{'}} = 1 \Rightarrow K (x) = x + λ \Rightarrow y (x) = \frac{x + λ}{\sqrt{1 + x^{2}}} (λ \in R)

Answered by john santu last updated on 27/Apr/20

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

I n t e g r a t i n g f a c t o r

\Rightarrow \int p (x) d x = \int \frac{x}{1 + x^{2}} d x

\int p (x) d x = \frac{1}{2} \int \frac{d (1 + x^{2})}{1 + x^{2}} = \ln \sqrt{1 + x^{2}}

u (x) = e^{\int p (x) d x} = \sqrt{1 + x^{2}}

s o l u t i o n

y = \frac{\int u (x) . q (x) d x + C}{u (x)}

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

y = \frac{x + C}{\sqrt{1 + x^{2}}} .