Find the function

**qzno** · February 18th 2009, 03:25 PM

Find a positive, continuous function f(x) which satisfies the integral equation

$f(x) = \pi \left( 2 + \int_{1}^{x} 3f(t) dt \right)$

Hint: start by differentiating each side.

I Have Absolutely NO Idea On How To Do This Problem, I Don't Even Understand The Hint.
Thanks To Anyone Who Helps <3
: )

**Reckoner** · February 18th 2009, 04:23 PM

Originally Posted by qzno

Find a positive, continuous function f(x) which satisfies the integral equation

$f(x) = \pi \left( 2 + \int_{1}^{x} 3f(t) dt \right)$

Hint: start by differentiating each side.

I Have Absolutely NO Idea On How To Do This Problem, I Don't Even Understand The Hint.
Thanks To Anyone Who Helps <3
: )

Differentiate both sides of the equation, as it says. The left side will be $f'(x),$ and for the right side, you will need to use the second part of the fundamental theorem of calculus. After that, you should be able to determine a suitable $f(x)$ through integration.

**qzno** · February 18th 2009, 04:32 PM

i still cant figure out how to differentiate the right side : (

**skeeter** · February 18th 2009, 04:34 PM

$\frac{d}{dx} \left[ \int_a^x f(t) \, dt \right] = f(x)$

**qzno** · February 18th 2009, 04:54 PM

i know that : (

**Reckoner** · February 18th 2009, 04:59 PM

Originally Posted by qzno

i know that : (

Okay, but we aren't mind readers. You need to tell us where you are getting stuck. Can you at least get the derivative? You have a constant (derivative is zero) plus an integral with a variable limit which can be differentiated using the property that skeeter gave; that part should be straightforward.

Show us your attempts so far, and we can guide you in the right direction.

**qzno** · February 19th 2009, 09:37 AM

I Got It Down To The Following:

$\int \frac{f'(x)}{f(x)} dx = \int 3\pi dx$

What do I do from here : )

**Reckoner** · February 19th 2009, 10:02 AM

Originally Posted by qzno

I Got It Down To The Following:

$\int \frac{f'(x)}{f(x)} dx = \int 3\pi dx$

What do I do from here : )

Good. The right side becomes $3\pi x + C\text.$ Now what is $\int\frac{u'}u\,dx?$

Apply the log rule, and then exponentiate both sides.

**qzno** · February 19th 2009, 10:05 AM

can you do u substitution and do like

u = f(x)
du = f'(x) dx

and then youd get

$\int \frac{1}{u} du$

**Reckoner** · February 19th 2009, 10:06 AM

Originally Posted by qzno

can you do u substitution and do like

u = f(x)
du = f'(x) dx

and then youd get

$\int \frac{1}{u} du$

Correct! You've got it.

**qzno** · February 19th 2009, 10:16 AM

so i got:

$ln (f(x)) + C = 3 \pi x + C$

is that all i do?

**Reckoner** · February 19th 2009, 10:22 AM

Originally Posted by qzno

so i got:

$ln (f(x)) + C = 3 \pi x + C$

Good, but the two constants of integration are not necessarily equal; you should either use separate names (like $C_0$ and $C_1$ ), or combine them as one constant.

is that all i do?

No. You want to find $f(x)$ (i.e., solve for $f(x)$ ).

Note that in this problem, you want to find a function that satisfies the conditions. There are many choices, depending on what you choose for $C\text.$ Just pick something that works.