1. ## Find the function

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

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

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
: )

2. Originally Posted by qzno
Find a positive, continuous function f(x) which satisfies the integral equation

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

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 $\displaystyle f'(x),$ 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 $\displaystyle f(x)$ through integration.

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

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

5. i know that : (

6. 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.

7. I Got It Down To The Following:

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

What do I do from here : )

8. Originally Posted by qzno
I Got It Down To The Following:

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

What do I do from here : )
Good. The right side becomes $\displaystyle 3\pi x + C\text.$ Now what is $\displaystyle \int\frac{u'}u\,dx?$

Apply the log rule, and then exponentiate both sides.

9. can you do u substitution and do like

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

and then youd get

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

10. Originally Posted by qzno
can you do u substitution and do like

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

and then youd get

$\displaystyle \int \frac{1}{u} du$
Correct! You've got it.

11. so i got:

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

is that all i do?

12. Originally Posted by qzno
so i got:

$\displaystyle ln (f(x)) + C = 3 \pi x + C$
Good, but the two constants of integration are not necessarily equal; you should either use separate names (like $\displaystyle C_0$ and $\displaystyle C_1$), or combine them as one constant.

is that all i do?
No. You want to find $\displaystyle f(x)$ (i.e., solve for $\displaystyle f(x)$).

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 $\displaystyle C\text.$ Just pick something that works.

13. $\displaystyle f(x) = e^{3 \pi x + c_2} - c_1$

or

$\displaystyle f(x) = e^{3 \pi x + c_2 - c_1}$

Are either of these correct?

14. Originally Posted by qzno
$\displaystyle f(x) = e^{3 \pi x + c_2 - c_1}$
Good. Now choose $\displaystyle c_1$ and $\displaystyle c_2$ so that $\displaystyle f$ satisfies the original equation.

15. how do u choose $\displaystyle c_1$ and $\displaystyle c_2$ ?