# Thread: What is this?

1. ## What is this?

A friend of mine gave this to me as a challenge. I almost have no clue what I'm supposed to do

he says "Solve for the following equation for f(x)"

$\displaystyle f(x) = x + \lambda \int_0^1 f(\xi) \,\,d \xi$

can anyone tell me what type of problem this is so I can do a bit of research on my own? My understanding of calculus only spans from Calc I to Calc II

2. ## Re: What is this?

Hey ReneG.

Hint: Try differentiating both sides to get a differential equation and solve from there.

(These kinds of problems are known as integro-differential equations)

Integro-differential equation - Wikipedia, the free encyclopedia

3. ## Re: What is this?

Thank you for pointing me in the right direction

4. ## Re: What is this?

Hi ReneG,
If the upper limit of your integral is x and not 1, I agree with Chiro. However as written, the integral is just a number c. So integrate both sides of your equation from 0 to 1 and get:

$\displaystyle c={1\over2}+\lambda c$ or $\displaystyle c={1\over2(1-\lambda)}$

So $\displaystyle f(x)=x+{\lambda\over2(1-\lambda)}$

5. ## Re: What is this?

Originally Posted by johng
Hi ReneG,
If the upper limit of your integral is x and not 1, I agree with Chiro. However as written, the integral is just a number c. So integrate both sides of your equation from 0 to 1
No typos. How did you integrate both sides though?

\displaystyle \begin{align*} f(x) &= x + \lambda \int_{0}^{1}f(\xi)\, d\xi \\ \int_{0}^{1}f(x)\,dx &= \int_0^1 x \,dx + \lambda \int_{0}^{1} \left[ \int_{0}^{1} f(\xi)\,d\xi \right ] \,dx \\ \int_{0}^{1}f(x)\,dx &= \frac{1}{2} + \lambda \int_{0}^{1} \left[ \int_{0}^{1} f(\xi)\,d\xi \right ]\,dx \end{align}

I'm lost.

6. ## Re: What is this?

Hi again,
It's easier for me to type the symbols in my own editor than to struggle with Latex and this HTML editor, so see the following "png" attachment:

I hope this clears it up.

7. ## Re: What is this?

Originally Posted by johng
I hope this clears it up.
I appreciate it, thank you.