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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Differential Equations

NDA 2021 · Mathematics · Differential Equations

A particle starts from origin with a velocity (in m / s ) given by...

A particle starts from origin with a velocity (in m/s\mathrm{m} / \mathrm{s} ) given by the equation dxdt=x+1\frac{d x}{d t}=x+1. The time (in second) taken by the particle to traverse a distance of 24m24 m is

Options

  1. Aln⁡24\ln 24
  2. Bln⁡5\ln 5
  3. C2ln⁡52 \ln 5
  4. D2ln⁡42 \ln 4

Correct answer

C. 2ln⁡52 \ln 5

Step-by-step solution

dxdt=x+1⇒dxx+1=dt\begin{aligned} & \frac{d x}{d t}=x+1 \\ & \Rightarrow \frac{d x}{x+1}=d t\end{aligned} Integrating both sides, we get ln⁡(x+1)=t+c\ln (x+1)=t+c. At t=0,x=0t=0, x=0 (origin point) Then, ln⁡(0+1)=0+c⇒c=0\ln (0+1)=0+c \Rightarrow c=0∴ln⁡(x+1)=twhen x=24,t=ln⁡(24+1)=ln⁡(25)=2ln⁡(5)\begin{aligned} & \therefore \ln (x+1)=t \\ & \text{when } x=24, t=\ln (24+1)=\ln (25)=2 \ln (5)\end{aligned}$$
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