A drug is infused into a patient's bloodstream at a constant rate of $r$ grams per second. Simultaneously, the drug is removed at a rate proportional to the amount $x(t)$ of the drug present at time $t$. Determine a differential equation for the amount $x(t)$. (Use $k > 0$ for the constant of proportionality and $x$ for $x(t)$.)

Neetesh Kumar | November 08, 2024

Differential Equation Homework Help

This is the solution to Math 2A, section 13Z, Fall 2023 | WebAssign
Math002ACh1Sec03 (Homework) Question - 9
Contact me if you need help with Homework, Assignments, Tutoring Sessions, or Exams for STEM subjects.
You can see our Testimonials or Vouches from here of the previous works I have done.

Get Homework Help

---

Step-by-Step-Solution:

Let $x(t)$ be the amount of the drug in the bloodstream at time $t$.

Step 1: Define the Rates

1. The rate at which the drug is infused into the bloodstream is a constant: $\text{Rate of infusion} = r \text{ grams/second}$

2. The rate at which the drug is removed from the bloodstream is proportional to the amount of drug present: $\text{Rate of removal} = k x(t)$

Where $k > 0$ is the constant of proportionality.

Step 2: Write the Differential Equation

The change in the amount of drug in the bloodstream over time, $\frac{dx}{dt}$, can be expressed as the difference between the rate of infusion and the rate of removal:

$\frac{dx}{dt} = r - k x(t)$

Conclusion

The differential equation governing the amount $x(t)$ of the drug in the bloodstream is:

$\frac{dx}{dt} = \boxed{r - k x}$

---

Please comment below if you find any error in this solution.
If this solution helps, then please share this with your friends.
Please subscribe to my Youtube channel for video solutions to similar questions.
Keep Smiling :-)