The particular solution of the differential equation \( \displaystyle\frac{dy}{dx}=y\tan x \); \( y=1 \) when \( x=0 \) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( y=secx \)
Separating the variables:
\( \displaystyle \frac{dy}{y}=\tan x\,dx \)
\( \displaystyle \log|y|=\log|\sec x|+\log|C| \)
\( \displaystyle y=C\sec x \)
Using the condition y = 1 when x = 0:
\( \displaystyle 1=C\sec 0=C \)
\( \displaystyle y=\sec x \)
Shortcut: Check both conditions in the options. At x = 0, only cos x and sec x give y = 1. For y = cos x, \( \frac{dy}{dx}=-\sin x \), while \( y\tan x=\sin x \), so it fails. For y = sec x, \( \frac{dy}{dx}=\sec x\tan x=y\tan x \), so it satisfies the equation.
NCERT (2026-27 edition):
Differential Equations with Variables Separable
