solve-the-de-x-2-1-y-xy-x-2-x-

Question Number 65402 by mathmax by abdo last updated on 29/Jul/19

s o l v e t h e (d e) \sqrt{x^{2} + 1} y^{^{″}} - {x y}^{^{'}} = x^{2} - x

Commented by mathmax by abdo last updated on 30/Jul/19

l e t y^{^{'}} = z (e) \Rightarrow \sqrt{x^{2} + 1} z^{^{'}} - x z = x^{2} - x

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

l n ∣ z ∣ = \sqrt{x^{2} + 1} + c \Rightarrow z = k e^{\sqrt{1 + x^{2}}}

m v c m e t h o d g i v e z^{^{'}} = k^{^{'}} e^{\sqrt{1 + x^{2}}} + k \frac{x}{\sqrt{1 + x^{2}}} e^{\sqrt{1 + x^{2}}}

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

\sqrt{1 + x^{2}} k^{^{'}} e^{\sqrt{1 + x^{2}}} + {k x e}^{\sqrt{1 + x^{2}}} - x k e^{\sqrt{1 + x^{2}}} = x^{2} - x \Rightarrow

k^{^{'}} = \frac{x^{2} - x}{\sqrt{1 + x^{2}}} e^{- \sqrt{1 + x^{2}}} \Rightarrow k (x) = \int \frac{x^{2} - x}{\sqrt{1 + x^{2}}} e^{- \sqrt{1 + x^{2}}} d x + c \Rightarrow

z (x) = (\int \frac{x^{2} - x}{\sqrt{1 + x^{2}}} e^{- \sqrt{1 + x^{2}}} d x + c) e^{\sqrt{1 + x^{2}}}

= e^{\sqrt{1 + x^{2}}} \int \frac{x^{2} - x}{\sqrt{1 + x^{2}}} e^{- \sqrt{1 + x^{2}}} d x + c e^{\sqrt{1 + x^{2}}}

y^{^{'}} = z (x) =\Rightarrow y (x) = \int^{x} z (t) d t + λ

= \int^{x} {e^{\sqrt{1 + t^{2}}} \int^{t} \frac{u^{2} - u}{\sqrt{1 + u^{2}}} e^{- \sqrt{1 + u^{2}}} d u + c e^{\sqrt{1 + t^{2}}}} d t + λ