The particular solution of $\dfrac{dy}{dx}=xe^{y-x}$, when $x=y=0$, is... (A) $e^{x-y}=x+1$ (B) $e^{x+y}=x+1$ (C) $e^x+e^y=x+1$ (D) $e^{y-x}=x-1$

The particular solution of (dy)/(dx)=xey-x, when x=y=0, is... (A) ex-y=x+1 (B) ex+y=x+1 (C) ex+ey=x+1 (D) ey-x

Miscellaneous Exercise 6(I) · Q107

\dfrac{dy}{dx}=xe^{y-x}

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$\dfrac{dy}{dx}=xe^{y-x}=xe^ye^{-x}$ separates as $e^{-y}dy=xe^{-x}dx$ . Integrating (RHS by parts): $-e^{-y}=-xe^{-x}-e^{-x}+c_1$ , i.e. $e^{-y}=(x+1)e^{-x}-c_1$ , i.e. $e^{x-y}=(x+1)-c_1e^x$ …

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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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Q.The particular solution of $\dfrac{dy}{dx}=xe^{y-x}$ , when $x=y=0$ , is... (A) $e^{x-y}=x+1$ (B) $e^{x+y}=x+1$ (C) $e^x+e^y=x+1$ (D) $e^{y-x}=x-1$

Concept understanding — Differential Equations with Variables Separable

Q.The particular solution of dydx=xey−x\dfrac{dy}{dx}=xe^{y-x}dxdy​=xey−x, when x=y=0x=y=0x=y=0, is... (A) ex−y=x+1e^{x-y}=x+1ex−y=x+1 (B) ex+y=x+1e^{x+y}=x+1ex+y=x+1 (C) ex+ey=x+1e^x+e^y=x+1ex+ey=x+1 (D) ey−x=x−1e^{y-x}=x-1ey−x=x−1

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Q.The particular solution of $\dfrac{dy}{dx}=xe^{y-x}$ , when $x=y=0$ , is... (A) $e^{x-y}=x+1$ (B) $e^{x+y}=x+1$ (C) $e^x+e^y=x+1$ (D) $e^{y-x}=x-1$

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