= Two-exponential-mixture survival probability
{title2=$\phi(u)=1-9e^{-u}/16-e^{-3u}/16$}
In the <classical risk model> with density $3e^{-4x}+e^{-2x}/2$ and <relative safety loading> $3/5$, the claim mean is $5/16$ and $\lambda/c=2$. The convolution terms $A_a(u)=\int_0^u\phi(u-x)e^{-ax}\,dx$ satisfy $A_a'=\phi-aA_a$. Applying $(D+4)(D+2)$ to $\phi'=2\phi-6A_4-A_2$ yields $\phi^{(3)}+4\phi''+3\phi'=0$. Its solution is fixed by $\phi(\infty)=1$, $\phi(0)=3/8$ and $\phi'(0)=3/4$, giving the displayed probability. The derivative condition comes from the original integral equation and excludes spurious solutions of the higher-order equation.
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