# Compound Interest Differential Equation Question

I'm positive someone else has asked a nearly identical problem in the past but I can't find an example of this specific variation.

Let's say initially you invest some principal amount ($$I_P$$) into a savings account with interest rate ($$r$$) continuously compounding interest. Additionally you invest a varying percentage ($$p(t)$$) of your wages ($$w(t)$$) such that your total investment into this account is given by:

$$I_{T}( t) =I_{P} +I_{w} ,\ \ \ \ \ \ \ \ I_{w} \ =\int p( t) w( t) \ dt$$

How would one set up a d.e. to determine the time value $$V(t)$$ of the investment? How would a variable interest rate $$r(t)$$ affect this?

At time $$T,$$ each amount that was invested in the interval $$[t,t+dt]$$ has compounded by $$e^{\int_t^Tr(s)ds}.$$ Therefore, taking the initial investment to have been at time $$t=0,$$ for each later time $$T,$$ $$V(T)=I_Pe^{\int_0^Tr(s)ds}+\int_{t=0}^Tp(t)w(t)e^{\int_t^Tr(s)ds}dt.$$