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GATE IN 2020 Official Paper

Option 1 : x(t) = 0

CT 1: Ratio and Proportion

2846

10 Questions
16 Marks
30 Mins

Given the differential equation is,

\(\frac{{dx}}{{dt}} = \sin \left( x \right)\)

\( \Rightarrow \frac{{dx}}{{\sin \left( x \right)}} = dt\)

By integrating on both the sides,

\(\smallint \frac{{dx}}{{\sin \left( x \right)}} = \;\smallint dt\)

log |cosec x – cot x| = t + c

Taking antilog on both sides, we get

⇒ cosec x – cot x = e^{t + c}

⇒ tan(x/2) = ke^{t}

⇒ x(t) = 2 tan^{-1}(ke^{t})

Now, putting the initial condition x(0) = 0 i.e. at t = 0, x = 0.

⇒ 0 = 2 tan^{-1}(ke^{t})

⇒ k = 0

Now, the solution becomes,

x(t) = 2 tan