Help me fast so that I will give good rating Find all solutions of differential equationxy" =1+ y2

Consider given differential equation put y” =t’ and y’ =t ⇒xt'=1+t2 ∫dt1+t2=∫dxx sinh-1t=logxc t…

Answered: plutions of differential equation ху"…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find all solutions of differential equation
xy" =
1+ y2

Expert Solution

Step 1

Consider given differential equation put y” =t’ and y’ =t

$\Rightarrow x t^{'} = \sqrt{1 + t^{2}}$

$\int \frac{d t}{\sqrt{1 + t^{2}}} = \int \frac{d x}{x}$

$\sin h^{- 1} t = \log x c$

t =sinh(logxc)

$\frac{d y}{d x} = \frac{e^{\log x c} - e^{- \log x c}}{2}$

$\frac{d y}{d x} = \frac{x c}{2} - \frac{1}{2 x c}$

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plutions of differential equation ху" 1+ y2