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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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