= Exponential-mixture differential equation for survival probability
For a <mixture distribution> of <exponential distributions>, introduce one <convolution> state $A_j(u)=\int_0^u\varphi(z)e^{-\beta_j(u-z)}\,dz$ for each rate $\beta_j$. It satisfies $A_j'=\varphi-\beta_jA_j$. Together with the <survival integro-differential equation> these form a constant-coefficient first-order system; applying the differential operators $D+\beta_j$ eliminates the <convolution> states. For equal mixing weights at rates one and one half, with $r=\lambda/c$, the result is $\varphi'''+(3/2-r)\varphi''+(1/2-3r/4)\varphi'=0$. The original integral equation supplies initial conditions lost in elimination.
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