Find the particular solution: $3e^x\tan y\cdot dx+(1+e^x)\sec^2y\cdot dy=0$, when $x=0,\ y=\pi$.

Find the particular solution: 3extan y· dx+(1+ex)sec²y· dy=0, when x=0,y=π.

Exercise 6.3 · Q43

3e^x\tan y\cdot dx+(1+e^x)\sec^2y\cdot dy=0

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$3e^x\tan y\,dx+(1+e^x)\sec^2y\,dy=0$ separates as $\dfrac{\sec^2y}{\tan y}dy=-\dfrac{3e^x}{1+e^x}dx$ . Integrating: $\log(\tan y)=-3\log(1+e^x)+c_1$ , i.e. $\tan y\,(1+e^x)^3=c$ . At $x=0,y=\pi$ : $\tan\pi=0$ , so the left side is $0\cdot(1+1)^3=0$ , forcing $c=0$ . Since $(1+e^x)^3$ is never zero, this forces $\tan y=0$ throughout — i.e. the initial point lies exactly on the singular branch $y=\pi$ (a constant f …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2024Set ANNUAL2 marks
$1+\dfrac{dy}{dx}=\csc(x+y)$
›Reveal solutionSolution

Put $u=x+y$ ; the equation becomes $\dfrac{du}{dx}=\csc u$ , separable.

$u=x+y \Rightarrow \dfrac{du}{dx}=1+\dfrac{dy}{dx}=\csc u$

$\sin u\,du=dx$

…

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CBSE 2023Set ANNUAL2 marksMCQ
$\dfrac{dx}{dt} = \dfrac{x\log x}{t}$
(a) $x = e^{ct}$
(b) $x + e^{ct} = 0$
(c) $x = e^t + t$
(d) $xe^{ct} = 0$
›Reveal solutionSolution

Separate variables and integrate.

$\dfrac{dx}{x\log x}=\dfrac{dt}{t}$

Integrating: $\log(\log x)=\log t+\log k=\log(kt)$

…

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CBSE 2023Set ANNUAL2 marks
$\cos x \cos y\,dy - \sin x \sin y\,dx = 0$
›Reveal solutionSolution

Separate variables: $\cot y\,dy=\tan x\,dx$ .

$\cos x\cos y\,dy=\sin x\sin y\,dx \Rightarrow \cot y\,dy=\tan x\,dx$

Integrating: $\log|\sin y|=-\log|\cos x|+\log k$

…

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CBSE 2022Set ANNUAL2 marks
$y\dfrac{dy}{dx} + x = 0$
›Reveal solutionSolution

Separate variables and integrate.

$y\dfrac{dy}{dx}+x=0 \implies y\,dy = -x\,dx$

…

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CBSE 2020Set ANNUAL2 marks
$\dfrac{dy}{dx} = x^2y + y$
›Reveal solutionSolution

Separate variables and integrate.

$\dfrac{dy}{dx} = y(x^2+1)$

Separating variables:

$\dfrac{dy}{y} = (x^2+1)\,dx$

Integrating both sides: …

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←Previous question · 6.3Q42 — Solve: (dy)/(dx)=ex+y+x²ey →Next question · 6.3Q44 — Find the particular solution: (x-y²x)· dx-(y+x²y)· dy=0, when x=2,y=0.

Q.Find the particular solution: $3e^x\tan y\cdot dx+(1+e^x)\sec^2y\cdot dy=0$ , when $x=0,\ y=\pi$ .

Concept understanding — Differential Equations with Variables Separable

Q.Find the particular solution: 3extan⁡y⋅dx+(1+ex)sec⁡2y⋅dy=03e^x\tan y\cdot dx+(1+e^x)\sec^2y\cdot dy=03extany⋅dx+(1+ex)sec2y⋅dy=0, when x=0, y=πx=0,\ y=\pix=0, y=π.

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Q.Find the particular solution: $3e^x\tan y\cdot dx+(1+e^x)\sec^2y\cdot dy=0$ , when $x=0,\ y=\pi$ .

Unlock everything free for 14 days