JEE Mains · Maths · STD 12 - 9. differential equations
Let \(x=x(t)\) and \(y=y(t)\) be solutions of the differential equations \(\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0\) and \(\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{by}=0\) respectively, \(\mathrm{a}, \mathrm{b} \in \mathrm{R}\). Given that \(x(0)=2 ; y(0)=1\) and \(3 y(1)=2 x(1)\), the value of \(t\), for which \(x(t)=y(t)\), is :
- A \(\log _{\frac{2}{3}} 2\)
- B \(\log _4 3\)
- C \(\log _3 4\)
- D \(\log _{\frac{4}{3}} 2\)
Answer & Solution
Correct Answer
(D) \(\log _{\frac{4}{3}} 2\)
Step-by-step Solution
Detailed explanation
\(\frac{d x}{d t}+a x=0 \) \( \frac{d x}{x}=-a d t \) \( \int \frac{d x}{x}=-a \int d t \) \( \ln |x|=-a t+c\) \( \text { at } t=0, x=2 \) \(\ln 2=0+c \) \( \ln x=-a t+\ln 2\) \( \frac{x}{2}=e^{-3 t}\) \( x=2 e^{-a t} \) \(............(i)\) \(\frac{d y}{d t}+b y=0 \)…
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