Give-an-example-of-differential-equation-which-has-no-solutions-

Question Number 3656 by prakash jain last updated on 17/Dec/15

Give an example of differential equation which

has no solutions .

Answered by Filup last updated on 18/Dec/15

W e i e r s t r a s s f u n c t i o n

is a fuction that is continuous at all

points but differentiable at none .

It is a real valued function .

f (x) = \underset{n = 0}{\sum^{\infty}} a^{n} \cos (b^{n} π x)

0 < a < 1

b \in O^{+} odd positive

a b > 1 + \frac{3}{2} π

(∴ b > 7)

Search wikipedia for 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

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

\frac{d y}{d x} = f (x) where \int f (x) d x does not exist

for any set A \subset R .

Commented by Filup last updated on 19/Dec/15

I^{'} m not sure there are any ?

What about :

\frac{d y}{d x} = x^{x^{x^{\dots}}}

y = \int x^{x^{x^{\dots}}} d x

? ? ?

Commented by Rasheed Soomro last updated on 19/Dec/15

I f w e d r a w a c u r v e w i t h a p e n c i l

b y h a n d i t m a y n o t h a v e a n e q u a t i o n

y e t i t m a y h a v e a r e a u n d e r i t w i t h i n

s p e c i f i e d v a l u e s o f x .

Commented by 123456 last updated on 19/Dec/15

f : [0, 1] \to R given by

f (x) = {\begin{cases} 1 & x \in Q \\ x & x \notin Q \end{cases}

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])

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