Differential equation: Substance in body I have the following problem: A substance is eaten by a person at a constant rate _a_ [milligrams per hour] . Let _y_ [milligrams] denote the amount of the substance in the body after _t_ hours, and let _b_ [milligrams] denote the starting amount of the substance in the body (i.e _y(0)=b_ ). The substance exits the body at a speed proportional to the current amount of the substance in the body. **1) Create a differential equation that describes this.** I've tried the following: 1) $ y=at-k\frac{dy} {dt} $ with the condition $y(0)=b$ I'm however not sure if this is the correct way to create an equation that describes the relation stated above, so feel free to correct me.

In a small time $dt$, the person will ingest $a\, dt$ of substance and will expel $k\, y\, dt$ of the substance. So $dy = a\, dt - k\, y\, dt$, or ${dy\over dt} = a - ky$.

Often when setting up differential equations, it helps to begin "In a small time $dt\dots$." Then construct the infinitesimal changes $dy$ in the dependent variable. Then divide through by $dt$ to form the differential equation.

Note that my "$k$" is different from yours (the units are different).