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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Differential Equations

NDA 2019 · Mathematics · Differential Equations

The differential equation which represents the family of curves...

The differential equation which represents the family of curves given by tan⁡y=c(1−ex)is\tan y=c \left(1-e^x\right) is

Options

  1. Aextan⁡ydx+(1−ex)dy=0e^x \tan y d x+\left(1-e^x\right) d y=0
  2. Bextan⁡ydx+(1−ex)sec⁡2ydy=0e^x \tan y d x+\left(1-e^x\right) \sec ^2 y d y=0
  3. Cex(1−ex)dx+tan⁡ydy=0e^x\left(1-e^x\right) d x+\tan y d y=0
  4. Dextan⁡ydy+(1−ex)dx=0e^x \tan y d y+\left(1-e^x\right) d x=0

Correct answer

B. extan⁡ydx+(1−ex)sec⁡2ydy=0e^x \tan y d x+\left(1-e^x\right) \sec ^2 y d y=0

Step-by-step solution

tan⁡y=c(1−ex)⇒sec⁡2ydydx=c(−ex)⇒c=1exsec⁡2ydydx∴tan⁡y=−1exsec⁡2y⋅dydx(1−ex)⇒ex⋅tan⁡y=sec⁡2y(1−ex)dydx⇒extan⁡ydx+(1−ex)sec⁡2ydy=0\begin{aligned} & \tan y=c\left(1-e^x\right) \\ & \Rightarrow \sec ^2 y \frac{d y}{d x}=c\left(-e^x\right) \\ & \Rightarrow c=\frac{1}{e^x} \sec ^2 y \frac{d y}{d x} \\ & \therefore \tan y=\frac{-1}{e^x} \sec ^2 y \cdot \frac{d y}{d x}\left(1-e^x\right) \\ & \Rightarrow e^x \cdot \tan y=\sec ^2 y\left(1-e^x\right) \frac{d y}{d x} \\ & \Rightarrow e^x \tan y d x+\left(1-e^x\right) \sec ^2 y d y=0\end{aligned}
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