The differential equation $y'(x)=y(x+a)$ does have solutions for $a=\frac\pi 2$ but what about other values of $a$? What about $a=1$?
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See this extensive solution of mine : https://math.stackexchange.com/questions/4611530/particular-differential-equation/4611566#4611566 – Abezhiko Jan 25 '23 at 17:11
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Did you try Laplace/Fourrier transform to solve it? – X0-user-0X Jan 25 '23 at 17:13