Given the integral equation: $$\int_0^a f(x)\left[ \frac{d^2}{dx^2}f(x) \right]dx=a$$ with the condition: $$\lim_{x\to\infty}f(x)=0$$ how can I find its solution? Is the solution (if any) the only one possible?
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$a$ is a constant or you wanna solve the OE w.r.t it? – Mikasa Jun 11 '13 at 10:43
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@BabakS.$a$ is a constant – Riccardo.Alestra Jun 11 '13 at 10:45
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Then what is the limit condition on $f$ doing there? This only makes sense if the equation is supposed to hold for every $a>0$. – Harald Hanche-Olsen Jun 11 '13 at 11:33
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@HaraldHanche-Olsen: I'm looking for a solution given a number $a\in\mathbb{R}$ – Riccardo.Alestra Jun 11 '13 at 11:54
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Okay, then my given answer is wrong. I'll delete it and add a new one. – Harald Hanche-Olsen Jun 11 '13 at 12:05
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Take any function $g$ defined on $[0,a]$, and write $$\int_0^a g(x)g''(x)\,dx=b.$$ All we require at this point is that $b>0$. Clearly, there is a huge number of such functions to choose from. Now define $$f(x)=\sqrt{\frac{a}{b}} g(x)\qquad\text{for } x\in[0,a],$$ and expand $f$ to $(a,\infty)$ in whatever way you like, just so long as $$\lim_{x\to\infty} f(x)=0.$$ Again, there is a huge range of possibilities.
Harald Hanche-Olsen
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This is not a great solution, but that is because the given problem also is not great (to put it mildly). – Harald Hanche-Olsen Jun 11 '13 at 12:10
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Just to expand on the previous point: The problem as stated asks for a function. A general function has infinitely many degrees of freedom. Then just two scalar conditions are given on this function, which reduces the degrees of freedom from infinity to infinity minus two. Which is also infinity. The problem just doesn't make a lot of sense. – Harald Hanche-Olsen Jun 11 '13 at 13:52