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Question Number 3656 by prakash jain last updated on 17/Dec/15
Give an example of differential equation which  has no solutions.
$$\mathrm{Give}\:\mathrm{an}\:\mathrm{example}\:\mathrm{of}\:\mathrm{differential}\:\mathrm{equation}\:\mathrm{which} \\ $$$$\mathrm{has}\:\mathrm{no}\:\mathrm{solutions}. \\ $$
Answered by Filup last updated on 18/Dec/15
Weierstrass function  is a fuction that is continuous at all  points but differentiable at none.  It is a real valued function.    f(x)=Σ_(n=0) ^∞ a^n cos(b^n πx)  0<a<1  b∈O^+     odd positive  ab>1+(3/2)π  (∴b>7)    Search wikipedia for more!!!
$${Weierstrass}\:{function} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{fuction}\:\mathrm{that}\:\mathrm{is}\:\mathrm{continuous}\:\mathrm{at}\:\mathrm{all} \\ $$$$\mathrm{points}\:\mathrm{but}\:\mathrm{differentiable}\:\mathrm{at}\:\mathrm{none}. \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{valued}\:\mathrm{function}. \\ $$$$ \\ $$$${f}\left({x}\right)=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{a}^{{n}} \mathrm{cos}\left({b}^{{n}} \pi{x}\right) \\ $$$$\mathrm{0}<{a}<\mathrm{1} \\ $$$${b}\in\mathbb{O}^{+} \:\:\:\:\mathrm{odd}\:\mathrm{positive} \\ $$$${ab}>\mathrm{1}+\frac{\mathrm{3}}{\mathrm{2}}\pi \\ $$$$\left(\therefore{b}>\mathrm{7}\right) \\ $$$$ \\ $$$$\mathrm{Search}\:\mathrm{wikipedia}\:\mathrm{for}\:\mathrm{more}!!! \\ $$
Commented by 123456 last updated on 18/Dec/15
theorem:  if a function is diferrentiable, then its  contonuous  conjecture:  if a function is cotinuous, then its diferentiable  the first is true and the second is false  in other way diferrentiability impy  continuity, but continuity dont imply  diferenciability  the function you give is just a example  on R of that fact
$$\mathrm{theorem}: \\ $$$$\mathrm{if}\:\mathrm{a}\:\mathrm{function}\:\mathrm{is}\:\mathrm{diferrentiable},\:\mathrm{then}\:\mathrm{its} \\ $$$$\mathrm{contonuous} \\ $$$$\mathrm{conjecture}: \\ $$$$\mathrm{if}\:\mathrm{a}\:\mathrm{function}\:\mathrm{is}\:\mathrm{cotinuous},\:\mathrm{then}\:\mathrm{its}\:\mathrm{diferentiable} \\ $$$$\mathrm{the}\:\mathrm{first}\:\mathrm{is}\:\mathrm{true}\:\mathrm{and}\:\mathrm{the}\:\mathrm{second}\:\mathrm{is}\:\mathrm{false} \\ $$$$\mathrm{in}\:\mathrm{other}\:\mathrm{way}\:\mathrm{diferrentiability}\:\mathrm{impy} \\ $$$$\mathrm{continuity},\:\mathrm{but}\:\mathrm{continuity}\:\mathrm{dont}\:\mathrm{imply} \\ $$$$\mathrm{diferenciability} \\ $$$$\mathrm{the}\:\mathrm{function}\:\mathrm{you}\:\mathrm{give}\:\mathrm{is}\:\mathrm{just}\:\mathrm{a}\:\mathrm{example} \\ $$$$\mathrm{on}\:\mathbb{R}\:\mathrm{of}\:\mathrm{that}\:\mathrm{fact} \\ $$
Commented by prakash jain last updated on 18/Dec/15
In the question I asked for an example of  differential equation for which no solution  exists. So the example could be  (dy/dx) =f(x) where ∫f(x)dx does not exist  for any set A⊂R.
$$\mathrm{In}\:\mathrm{the}\:\mathrm{question}\:\mathrm{I}\:\mathrm{asked}\:\mathrm{for}\:\mathrm{an}\:\mathrm{example}\:\mathrm{of} \\ $$$$\mathrm{differential}\:\mathrm{equation}\:\mathrm{for}\:\mathrm{which}\:\mathrm{no}\:\mathrm{solution} \\ $$$$\mathrm{exists}.\:\mathrm{So}\:\mathrm{the}\:\mathrm{example}\:\mathrm{could}\:\mathrm{be} \\ $$$$\frac{\mathrm{d}{y}}{\mathrm{d}{x}}\:={f}\left({x}\right)\:\mathrm{where}\:\int{f}\left({x}\right){dx}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist} \\ $$$$\mathrm{for}\:\mathrm{any}\:\mathrm{set}\:\mathrm{A}\subset\mathbb{R}. \\ $$
Commented by Filup last updated on 19/Dec/15
I′m not sure there are any?    What about:  (dy/dx)=x^x^x^(...)     y=∫x^x^x^(...)   dx  ???
$$\mathrm{I}'\mathrm{m}\:\mathrm{not}\:\mathrm{sure}\:\mathrm{there}\:\mathrm{are}\:\mathrm{any}? \\ $$$$ \\ $$$$\mathrm{What}\:\mathrm{about}: \\ $$$$\frac{{dy}}{{dx}}={x}^{{x}^{{x}^{…} } } \\ $$$${y}=\int{x}^{{x}^{{x}^{…} } } {dx} \\ $$$$??? \\ $$
Commented by Rasheed Soomro last updated on 19/Dec/15
If  we draw a curve with a pencil  by hand it may not have an equation  yet it may have area under it within  specified values of x.
$$\mathcal{I}{f}\:\:{we}\:{draw}\:{a}\:{curve}\:{with}\:{a}\:{pencil} \\ $$$${by}\:{hand}\:{it}\:{may}\:{not}\:{have}\:{an}\:{equation} \\ $$$${yet}\:{it}\:{may}\:{have}\:{area}\:{under}\:{it}\:{within} \\ $$$${specified}\:{values}\:{of}\:{x}. \\ $$
Commented by 123456 last updated on 19/Dec/15
f:[0,1]→R given by  f(x)= { (1,(x∈Q)),(x,(x∉Q)) :}  if you plot f, it must be two line since  Q and I are denses into R, this function  is not riemann integrable, however its  lesbegue integrable (if im dont wrong  I was a 0 lesbegue measure, so the integral  may return 1 over [0,1])
$${f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R}\:\mathrm{given}\:\mathrm{by} \\ $$$${f}\left({x}\right)=\begin{cases}{\mathrm{1}}&{{x}\in\mathbb{Q}}\\{{x}}&{{x}\notin\mathbb{Q}}\end{cases} \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{plot}\:{f},\:\mathrm{it}\:\mathrm{must}\:\mathrm{be}\:\mathrm{two}\:\mathrm{line}\:\mathrm{since} \\ $$$$\mathbb{Q}\:\mathrm{and}\:\mathbb{I}\:\mathrm{are}\:\mathrm{denses}\:\mathrm{into}\:\mathbb{R},\:\mathrm{this}\:\mathrm{function} \\ $$$$\mathrm{is}\:\mathrm{not}\:\mathrm{riemann}\:\mathrm{integrable},\:\mathrm{however}\:\mathrm{its} \\ $$$$\mathrm{lesbegue}\:\mathrm{integrable}\:\left(\mathrm{if}\:\mathrm{im}\:\mathrm{dont}\:\mathrm{wrong}\right. \\ $$$$\mathbb{I}\:\mathrm{was}\:\mathrm{a}\:\mathrm{0}\:\mathrm{lesbegue}\:\mathrm{measure},\:\mathrm{so}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\left.\mathrm{may}\:\mathrm{return}\:\mathrm{1}\:\mathrm{over}\:\left[\mathrm{0},\mathrm{1}\right]\right) \\ $$

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