Exercise · ZW-EXPONENTIAL-GROWTH-AND-DECAY-002 · ★☆☆ Basic
Solve \(y'=-\tfrac12 y\) with \(y(0)=4\).
Choose one option.
Show answer
- C. \(y=4e^{-t/2}\)
Explanation
With \(k=-\tfrac12<0\) there is decay: \(y=4e^{-t/2}\).
Review the concept: Exponential growth and decay
A quantity \(y\) with initial value \(y_0\) grows or decays exponentially when its rate of change is proportional to itself, with constant \(k\): it grows if \(k>0\), decays if \(k<0\).
\[y'=ky\;\Longrightarrow\; y(t)=y_0\,e^{kt}\]
More practice problems: exponential growth and decay
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-001 Solve y'=3y with y(0)=2.
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-003 A population satisfies y'=ky, with y(0)=100 and y(2)=200. What is y(6)?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-004 A substance has a half-life of 5 years and satisfies y'=ky. What is k?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-005 A capital grows with y'=0.04y. After roughly how long does it double?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-006 In y'=ky with k<0 and y(0)>0, what happens to y(t) as t→∞?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-007 A 100 mg isotope keeps 80 mg after 3 years. With exponential decay, how many mg remain after 9 year…
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-008 With y'=0.1y, about when does the quantity triple?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-009 If y'=ky with y(1)=6 and y(3)=24, what is y(0)?
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