Solve: $\dfrac{\cos^2y\cdot dy}{x}+\dfrac{\cos^2x\cdot dx}{y}=0$

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$\dfrac{\cos^2y\,dy}{x}+\dfrac{\cos^2x\,dx}{y}=0$ . Multiplying through by $xy$ separates it: $y\cos^2y\,dy=-x\cos^2x\,dx$ , i.e. $y\cos^2y\,dy+x\cos^2x\,dx=0$ . Using $\cos^2\theta=\tfrac{1+\cos2\theta}{2}$ and integrating by parts, $\int t\cos^2t\,dt=\dfrac{t^2}{4}+\dfrac{t\sin2t}{4}+\dfrac{\cos2t}{8}+c$ . Applying this to both the $x$ - and $y$ -integrals and combining constants: $\dfrac{x^2}{4}+\dfrac{x\sin2x}{4}+\dfrac{\co …

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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.Solve: $\dfrac{\cos^2y\cdot dy}{x}+\dfrac{\cos^2x\cdot dx}{y}=0$

Concept understanding — Differential Equations with Variables Separable

Q.Solve: cos⁡2y⋅dyx+cos⁡2x⋅dxy=0\dfrac{\cos^2y\cdot dy}{x}+\dfrac{\cos^2x\cdot dx}{y}=0xcos2y⋅dy​+ycos2x⋅dx​=0

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Q.Solve: $\dfrac{\cos^2y\cdot dy}{x}+\dfrac{\cos^2x\cdot dx}{y}=0$

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