Find the general solution of this ODE dxI+ ylz

Given differential equation is ddxyy'(y')2+1-(y')2+1=0 (1) Since we know that…

Answered: %3D I+ ylz

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 the general solution of this ODE

Expert Solution

Step 1

Given differential equation is

$\frac{d}{d x} (\frac{y y'}{\sqrt{{(y')}^{2} + 1}}) - \sqrt{{(y')}^{2} + 1} = 0$ (1)

Since we know that

$\frac{d}{d x} (\frac{f (x)}{g (x)}) = \frac{g (x) f' (x) - f (x) g' (x)}{(g {(x))}^{2}}$

using above formula, (1) will be

$\frac{\sqrt{{(y')}^{2} + 1} [{(y')}^{2} + y y''] - y y' (\frac{2 y' y''}{2 \sqrt{{(y')}^{2} + 1}})}{{(y')}^{2} + 1} - \sqrt{{(y')}^{2} + 1} = 0 \Rightarrow \frac{({(y')}^{2} + 1) ({(y')}^{2} + y y'') - y y'^{2} y''}{{({(y')}^{2} + 1)}^{\frac{3}{2}}} - \sqrt{{(y')}^{2} + 1} = 0 \Rightarrow \frac{{(y')}^{2} + y y'' + {(y')}^{4} + y {(y')}^{2} y'' - y y'^{2} y'' - (1 + y'^{4} + 2 y'^{2})}{{({(y')}^{2} + 1)}^{\frac{3}{2}}} = 0 \Rightarrow y y'' - y'^{2} - 1 = 0$