# Find a differentiable function with the given properties

• Apr 15th 2009, 01:30 PM
linda2005
Find a differentiable function with the given properties
hello, need your help

find all function $\displaystyle f$ differentiable over $\displaystyle \mathbb{R}$
$\displaystyle - (\forall x\in \mathbb{R}) : \big(f'(x)\big)^2-\big(f(x)\big)^2=1 \quad (1)$
$\displaystyle -f'(0)=1 \quad(2)$
$\displaystyle -f'$ is differentiable over $\displaystyle \mathbb{R}\quad (3)$
• Apr 15th 2009, 02:06 PM
Calculus26
Ok not to be a pain but the word is differentiable not derivable.

Taking the square root

f ' ^2 -f = 1 or -1

Differentiate -- in either case

2f " f ' -f ' = 0

f ' (2f"-1) = 0

Now solve f ' = 0 (use original equation to find values of c) and 2f " - 1 = 0

That should be enough to get you to the end
• Apr 15th 2009, 02:13 PM
Jhevon
Quote:

Originally Posted by Calculus26
Ok not to be a pain but the word is differentiable not derivable.

Taking the square root

f ' ^2 -f = 1 or -1

either i'm missing something or you made a mistake here
• Apr 15th 2009, 02:31 PM
linda2005
Calculus26 it's $\displaystyle \big(f'(x)\big)^2-\big(f(x)\big)^2=1$ and not $\displaystyle \big(f'(x)\big)^2-f(x)=1$
• Apr 15th 2009, 02:44 PM
Calculus26
Sorry I read it too quickly I was seeing

[f '^2 -f]^2 = 1 for the original

For (f ')^2 - f^2 =1

the process is still the same

2f ' f "- 2 f f '= 0

f ' = 0 or f '' -f = 0

yields f = c from which there are no solutions

or f " -f = 0 which yield Acos(x) + Bsin(x) now apply f '(0) = 1

and finish
• Apr 15th 2009, 03:09 PM
Calculus26
I believe there are no solutions as f ' (0) =1 yields f =Acos(x) +sin(x)

Plugging this into the original there are no values of A Solving the equation.
• Apr 15th 2009, 03:20 PM
linda2005
Quote:

Originally Posted by Calculus26
or f " - f = 0 which yield Acos(x) + Bsin(x) now apply f '(0) = 1

sorry but you are wrong , you mean

f " + f = 0 which yield Acos(x) + Bsin(x)

but here we have f " - f = 0
• Apr 15th 2009, 03:26 PM
Jhevon
Quote:

Originally Posted by linda2005
sorry but you are wrong , you mean

f " + f = 0 which yield Acos(x) + Bsin(x)

but here we have f " - f = 0

yes, take $\displaystyle f(x) = Ae^x + Be^{-x}$ instead
• Apr 15th 2009, 03:29 PM
Calculus26
Ok I've made every possible mistake on this to now--thats what i get for hurrying

we get f = Ae^t + Be^-t

f ' (0) yields A+B =1

Substituting into original equation yields -4AB =1

Now solve --solutions are when A = (2^1/2+1)/2 or (1-2^1/2)/2

with corresponding values for B = 1 -A

Sorry for the mistakes
• Apr 15th 2009, 03:37 PM
Jhevon
Quote:

Originally Posted by Calculus26
Ok I've made every possible mistake on this to now

i'm sure you can find more mistakes to make if you tried ;)

Quote:

Sorry for the mistakes
it happens to the best of us, don't worry about it. mistakes were corrected before they caused much damage.

did you double check your solutions for A? :D

i'm not going to check them, i trust you (Nod)
• Apr 15th 2009, 03:47 PM
Calculus26
I rechecked for both A and B in the original both by hand and with Mathcad to make sure-- thanx for the kind words

But still I always told my students to slow down --Calculus is more about patience than intelligence I'd say--too bad I don't follow my own advice.

That's what happens a year removed from teaching--skills are slipping
• Apr 15th 2009, 04:15 PM
linda2005
I find $\displaystyle f(x) = \frac{e^x - e^{-x}}{2}$ I'm wrong or you are Calculus26:D( f ' (0) yields A+B =1 false we have f ' (0) yields A-B =1) thanks you guys xxx
• Apr 15th 2009, 04:34 PM
Calculus26
Damn--- you're right using the initial condition A - B = 1 not A+B =1

as I said before--my solution satisfies the equation which is where I checked my results but not the initial conditions.

Now I quit

3 mistakes on one problem
• Apr 15th 2009, 04:50 PM
Calculus26
By the way not that it matters that much but

http://www.mathhelpforum.com/math-he...5c0cddb6-1.gif is also known as the hyperbolic sine and written

http://www.mathhelpforum.com/math-he...5c0cddb6-1.gif = sinh(x)

You may see the answer written that way

Thanx for your kindness amongst my blunders I should have stopped after

describing the process.
• Apr 15th 2009, 04:54 PM
Jhevon
Quote:

Originally Posted by Calculus26
Thanx for your kindness amongst my blunders I should have stopped after

describing the process.

i told you you could find more mistakes if you tried :D

that was mean. i am only messing with you Calc. but yes, if you find you made 2 blunders, just resort to describing the process and letting the OP do the groundwork. this comes with experience on forums (Nod)

take care (Handshake)