Continuously compounded interest with additional monthly deposits What is the equation for a continuously compounded with monthly additions of $300$ dollars for the first $10$ years and $500$ for the next $20$ with an initial investment of $0$?
I know the equation $Pe^{rt}$ is used but i don't understand how to set it so $300$ dollars is added every month.

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*Let $C_0$ the initial investment (at time 10).

*Let $C_1=300$ be the monthly investment compounded continuously at a nominal rate $\delta$ during the first 10 years

*Let $C_2=500$ be the monthly investment compound continuously at a nominal rate $\delta$ for the following 20 years after the initial 10 year period.

The present value of the total investment is given by
$$PV_I= C_0 + C_1\int^{10}_{0} e^{-\delta t}\,dt + C_2\int^{30}_{10} e^{-\delta t}\,dt C_0+ C_1\frac{1-e^{-10\delta}}{\delta}+ C_2\frac{e^{-10\delta}-e^{-30\delta}}{\delta}$$
If monthly investment of $C_1$ is done immediately after the initial deposit
$$PV_A= C_0 + C_1\int^{10}_{1/12} e^{-\delta t}\,dt + C_2\int^{30}_{10} e^{-\delta t}\,dt= C_0+ C_1\frac{e^{-\delta/12}-e^{-10\delta}}{\delta}+ C_2\frac{e^{-10\delta}-e^{-30\delta}}{\delta}$$
If the monthly investment of $C_2$ is done one month after the initial investment of $C_0$.
