- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Slope fields and Euler’s method practice question 2
A drug is given by a steady intravenous drip and cleared by the body at a rate proportional to the amount present, so the amount , in mg after hours, satisfies . For exactly one value of the constant , the function is a solution for every constant . Find .
How do you check that a given function solves a differential equation? Do that with this , keeping and as letters, and see what it demands of .
Differentiate , then substitute all of into . The two sides must agree for every and every .
A number.
Theory UBC Math 100 §11.2Key results
Worked solution
Strategy
To check a proposed solution, substitute it into both sides of the equation and compare. The exponential terms match for any C; the constant terms match for one value of b only.
Working
The constant b differentiates to 0, and the chain rule brings down −0.4.
Substitute the whole of Q, constant term included.
The exponential terms cancel and leave a condition on b alone. At 20 mg, clearance of 0.4 × 20 = 8 mg/h exactly balances the drip.
Answer
The answer is b equals 20 milligrams.
Check
At Q = 20 mg the drip of 8 mg/h exactly balances clearance of 0.4 × 20 = 8 mg/h, so the amount holds steady there.
Where this goes wrong
Substituting only part of Q into the right-hand side, or losing the 8, so that the constant terms are never compared. Every term of the proposed solution goes into both sides.
Leans on
The derivative of ekt, from The chain rule and logarithms
Try a similar question
Check that solves for every positive constant . Then find the solution with , and give .Saved in this browser only.