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I would like to solve this differential equation that is similar to the delayed differential equation here.

this is the DE: $f'(x) = f(kx), \;$ for some real $k$.

The reason for attempting this problem is that I am interested in solving this DE but do not have the means to solve it:

$øf(øx) = 2f(x)∫f(t)\mathrm{d}t$ integrated over $[0,x]$.

I though I'd start with a simpler yet similar problem to help me understand the original question.

amWhy
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Mathew
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  • Haven't you studied Laplace Transform ? It's the ideal tool for this kind of questions. – Jean Marie Apr 02 '17 at 20:45
  • The title does not fit what you are asking ! – Jean Marie Apr 02 '17 at 20:46
  • I realize that delayed differential equations refers to equations involving f(t) and f(t+k) and their derivatives but I'm not sure what to call this sort of thing when it is scaled by a factor rather than shifted. Also, I don't see how laplace transforms would help, I did try that method but got this for the first equation: s L{f(t)} - f(0) = L{f(kt)} (L should be a scripted L but I cant type that) I'm left with a problem I can't solve. Thanks for the power series solution @RobertIsrael, very helpful. – Mathew Apr 03 '17 at 21:18

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If you write formally $f(x) = \sum_{i=0}^\infty a_i x^i$, your equation $f'(x) = f(kx)$ becomes $$ \sum_{i=0}^\infty ((i+1) a_{i+1} - k^i a_i) x^i$$ The solution to the recurrence $(i+1) a_{i+1} = k^i a_i$ is $$ a_i = a_0 \frac{k^{i(i-1)/2}}{i!}$$ The series $$ \sum_{i=0}^\infty \frac{k^{i(i-1)/2}}{i!} x^i$$ converges to an entire function of $x$ for $|k| < 1$.

Robert Israel
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