Differential Equations
Exponential Decay — Finding Time
nta_pyq_2024_jan
Grade 12
Question:
Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\dfrac{dx}{dt}+ax=0$ and $\dfrac{dy}{dt}+by=0$ respectively, $a,b\in\mathbb{R}$. Given that $x(0)=2$; $y(0)=1$ and $3y(1)=2x(1)$, the value of $t$, for which $x(t)=y(t)$, is:
$\log_{\frac{2}{3}}2$
$\log_4 3$
$\log_3 4$
$\log_{\frac{4}{3}}2$
Step-by-Step Solution
Key Concept: Solve each DE: $x=2e^{-at}$, $y=e^{-bt}$. Use $3y(1)=2x(1)$ to find relation between $a$ and $b$. Then solve $x(t)=y(t)$ for $t$.
$x=2e^{-at}$, $y=e^{-bt}$. $3e^{-b}=4e^{-a}\Rightarrow e^{a-b}=\frac{4}{3}$. $x(t)=y(t)\Rightarrow2=e^{(a-b)t}=\left(\frac{4}{3}\right)^t\Rightarrow t=\log_{4/3}2$.
Correct Answer: 4