Find the particular solution: $y(1+\log x)=(\log x^x)\dfrac{dy}{dx}$, when $y(e)=e^2$.

Find the particular solution: y(1+log x)=(log xx)(dy)/(dx), when y(e)=e².

Miscellaneous Exercise 6(II) · Q138

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

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$y(1+\log x)=(\log x^x)\dfrac{dy}{dx}$ , and $\log x^x=x\log x$ , so $\dfrac{dy}{dx}=\dfrac{y(1+\log x)}{x\log x}$ — exactly the reciprocal of Exercise 6.3 Q3(iii)'s equation, with the same initial condition. So (as solved there) $y=e\ …

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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.Find the particular solution: $y(1+\log x)=(\log x^x)\dfrac{dy}{dx}$ , when $y(e)=e^2$ .

Concept understanding — Differential Equations with Variables Separable

Q.Find the particular solution: y(1+log⁡x)=(log⁡xx)dydxy(1+\log x)=(\log x^x)\dfrac{dy}{dx}y(1+logx)=(logxx)dxdy​, when y(e)=e2y(e)=e^2y(e)=e2.

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Q.Find the particular solution: $y(1+\log x)=(\log x^x)\dfrac{dy}{dx}$ , when $y(e)=e^2$ .

Unlock everything free for 14 days