solve-1-x-2-2-y-2x-1-x-2-y-2-0-

Question Number 103412 by abdomsup last updated on 14/Jul/20

s o l v e {(1 + x^{2})}^{2} y^{^{″}} + 2 x (1 + x^{2}) y^{^{'}} + 2 = 0

Answered by OlafThorendsen last updated on 15/Jul/20

{(1 + x^{2})}^{2} y^{^{″}} + 2 x (1 + x^{2}) y^{^{'}} + 2 = 0

y^{'} = \frac{u}{1 + x^{2}}

\Rightarrow y^{″} = \frac{(1 + x^{2}) u^{'} - 2 x u}{{(1 + x^{2})}^{2}}

(1 + x^{2}) u^{'} - 2 x u + 2 x u + 2 = 0

u^{'} = - \frac{2}{1 + x^{2}}

\Rightarrow u = - 2 Arctan x + C_{1}

y^{'} = - 2 \frac{Arctan x}{1 + x^{2}} + \frac{C_{1}}{1 + x^{2}}

\Rightarrow y = - {Arctan}^{2} x + C_{1} Arctan x + C_{2}

Commented by mathmax by abdo last updated on 15/Jul/20

thank you sir