# A particle starts from origin with a velocity (in m/s) given by the equation $\rm \frac{dx}{dt}=x+1$. The time (in seconds) taken by the particle to

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A particle starts from origin with a velocity (in m/s) given by the equation $\rm \frac{dx}{dt}=x+1$. The time (in seconds) taken by the particle to traverse a distance of 24 m is:
1. ln 24
2. ln 5
3. 2 ln 5
4. 2 ln 4

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Correct Answer - Option 3 : 2 ln 5

Formula Used:

$\int \frac {1}{x} dx = lnx + c$

At t = 0, Distance covered (x) = 0

Distance covered by the particle (x) = 24 m

Calculation:

The given differential equation $\rm \frac{dx}{dt}=x+1$ is in variable separable form.

On separating the variables, we get

⇒ $\rm \left(\frac{1}{x+1}\right)dx = dt$

On integrating both sides, we get:

⇒ ln (x + 1) = t + C    ---(i)

According to the question at t = 0, x = 0.

ln (0 + 1) = 0 + C

⇒ C = 0

Now, From (i), we get

⇒ ln (x + 1) = t

Again according to the question

⇒ t = ln (24 + 1)

⇒ t = ln 25

⇒ t = ln 52

∴ The required time is 2 ln 5.