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Solve the following integral equation:

$\phi(x)+\displaystyle \int^1_0 xt(x+t)\phi(t)\,dt=x $ , $0 \le x \le 1 $

I need to solve the integral equation above. Can anyone help me please?

Amzoti
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sumon
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2 Answers2

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Hint: $$\phi(x)+x^2 \displaystyle \int_0^1t\phi(t)dt+x\int_0^1t^2\phi(t)=x$$ i.e. $$\tag{1}\phi(x)=c_1x^2+c_2x $$

Where $c_1 = - \displaystyle \int_0^1t\phi(t)dt \ \ $ and $\ \ c_2=1- \displaystyle \int_0^1t^2\phi(t)dt$

Put $\phi(t)$ from $(1)$ in these integrals and calculate $c_1 \ , \ c_2$ from two equations you get

(two-equations two variables).

Pedro
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ABC
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  • You can use the \tag{} command to tag equations. Apparently, it doesn't accept \spadesuit as an input, sorry about that. – Pedro May 17 '13 at 01:37
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Related problems: (I). Here is a detailed solution that maybe someone benefits from it. Rearranging the equation as $$ \phi(x)= x- \displaystyle \int^1_0 xt(x+t)\phi(t)\,dt \longrightarrow (1). $$

$$ \phi(x) = x - x^2\int_{0}^{1}t\phi(t)\,dt - x \int_{0}^{1}t^2\phi(t) \,dt $$

$$ \implies \phi(x) = x - x^2 c_1 - x c_2 \longrightarrow (2).$$

Now, just subs back in the integral equation $(1)$ and compare the coefficients of $x's$, you will get a system of 2 equations in $c_1$ and $c_2$

$$ \frac{5}{4}c_{1}+\frac{1}{3}c_{2}= \frac{1}{3} $$

$$ \frac{1}{5}c_{1}+\frac{5}{4}c_{2}= \frac{1}{4}. $$

Solving the above system gives

$$ \left\{ c_{{1}}={\frac {80}{359}},c_{{2}}={\frac {59}{359}} \right\}. $$

Subs back in $(2)$ yields the solution

$$ \phi \left( x \right) ={\frac {300}{359}}\,x-{\frac {80}{359}}\,{x}^{2 },$$

which can be checked by plugging back in $(1)$.

Community
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Mhenni Benghorbal
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  • @moderators: This down vote is a misleading. – Mhenni Benghorbal May 16 '13 at 16:22
  • @Downvoters: What's the down vote for? This is the second one. – Mhenni Benghorbal May 17 '13 at 01:45
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    This solution is correct, +1. I don't know why it was downvoted. But does @moderators really notify the moderators? And do they really have to be notified in such a case? What could they do anyway? – Julien May 17 '13 at 01:45
  • @julien: It is really a misleading down vote. I worked out the problem in details so it will be like a reference for people. – Mhenni Benghorbal May 17 '13 at 01:48
  • Also, note that instead of substituting back and comparing coefficients, you can do $c_1=\int t\phi(t)=(1-c_2)\int t^2-c_1\int t^3$ and likewise for $c_2=\int t^2\phi(t)$. – Julien May 17 '13 at 01:48
  • @julien: You are right. Verifying the answer is a good exercise for the OP. – Mhenni Benghorbal May 17 '13 at 01:52
  • You understood that I upvoted, right? I even took the time to check the computations up to the system. Which is why I don't understand the downvotes either. But again, what do you expect the moderators to do about that? – Julien May 17 '13 at 01:52
  • @julien: We need to find a solution about these misleading down votes or those ones that happened by mistakes. However, whenever this happens, I usually put a comment under my answers and the post of the OP. – Mhenni Benghorbal May 17 '13 at 01:55
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    And what is the outcome of doing so? Has a moderator ever done anything about it? Are they notified by @moderators? What can they do anyway? I think you should ask yourself these questions first. – Julien May 17 '13 at 01:56
  • @julien: You are wright. One time, I found a comment says "why moderators have to be notified about that?"!!. So, I thought they should be notified. – Mhenni Benghorbal May 17 '13 at 02:02