xy" -y'=ohasan in finilereumber f solubons. Howeversahisfiesలోsoluhionthesenoinitialcondlih'onsy co)=0, y'co))don't thisresult contiaidichwhythe uniqueness theorem ?

Answered: xy" -yl=o has an infinite of soluhons.…

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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Question

xy" -y'=o
has
an in finile
reumber f solubons. However
sahisfies
లో
soluhion
these
no
initial
condlih'ons
y co)=0, y'co))
don't this
result contiaidich
why
the uniqueness theorem ?

Expert Solution

Step 1

Let us solve the given differential equation $x y^{''} - y^{'} = 0$

For the second order Euler homogeneous ODE of the form $a x^{2} y^{''} + b x y^{'} + c y = 0$

The solution will be of the form $y = x^{r}$

Putting

$y = x^{r} y^{'} = r x^{r - 1} y^{''} = r (r - 1) x^{r - 2}$

in the given differential equation we get,

$r (r - 1) x^{r - 1} - r x^{r - 1} = 0 x^{r - 1} (r^{2} - r - r) = 0 x^{r - 1} r (r - 2) = 0 E i t h e r r = 0 o r r = 2$

since they are distinct and real so the general solution of the given differential equation is

$y = C_{1} x^{2} + C_{2} x^{0} y = C_{1} x^{2} + C_{2}$

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xy" -yl=o has an infinite of soluhons. However reimber soluhion sahisfies these y co) =O, y'cool initial conditions don't thi's result contiadict why the uniqueness ther em ?