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The commenters got it almost solved, missing only that $t_{\text{min}}$ is an unknown instead of a parameter. The question should be: for what values of $t_{\text{min}}$ does the integral equation have a nontrivial solution? As shown in the comments, the function satisfies $$ [2 + E(t)] f(t) + 3 f'(t) + f''(t) = 0\\ 2 f(0) + f'(0) = 0\\ f(t_{\text{min}}) + ...


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