JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y = y(x)\) be the solution of the differential equation \(\dfrac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)\), \(y(0) = \dfrac{1}{2}\). Then \((2y(1) - 1)\) is equal to:
- A \(\sqrt{3}\tan\left(\dfrac{11\sqrt{3}}{6}\right)\)
- B \(\dfrac{\sqrt{3}}{2}\tan\left(\dfrac{11\sqrt{3}}{12}\right)\)
- C \(\sqrt{3}\tan\left(\dfrac{11\sqrt{3}}{12}\right)\)
- D \(\dfrac{\sqrt{3}}{2}\tan\left(\dfrac{11\sqrt{3}}{6}\right)\)
Answer & Solution
Correct Answer
(C) \(\sqrt{3}\tan\left(\dfrac{11\sqrt{3}}{12}\right)\)
Step-by-step Solution
Detailed explanation
The given differential equation is \(\dfrac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)\). Separating the variables: \(\dfrac{dy}{y^2 - y + 1} = (x^2 + x + 1) dx\) Integrating both sides: \(\int \dfrac{dy}{\left(y - \dfrac{1}{2}\right)^2 + \dfrac{3}{4}} = \int (x^2 + x + 1) dx\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- The number of real values \(\lambda\), such that the system of linear equations \(2 x-3 y+5 z=9\) ; \(x+3 y-z=-18\) ; \(3 x-y+\left(\lambda^{2}-1 \lambda \mid\right) z=16\) has no solution, is :-JEE Mains 2022 Hard
- Let \(X\) be a set containing \(10\) elements and \(P(X)\) be its power set. If \(A\) and \(B\) are picked up at random from \(P(X),\) with replacement, then the probability that \(A\) and \(B\) have equal number elements, isJEE Mains 2015 Hard
- An urn contains \(6\) white and \(9\) black balls. Two successive draws of \(4\) balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :JEE Mains 2024 Medium
- Let \(\vec{a}, \vec{b}, \vec{c}\) be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle \(\theta\), with the vector \(\vec{a}+\vec{b}+\vec{c}\). Then \(36 \cos ^{2} 2 \theta\) is equal to \(.....\)JEE Mains 2021 Hard
- Let \(f: R \rightarrow R\) be a differentiable function that satisfies the relation \(f ( x + y )= f ( x )+ f ( y )-1, \forall x\), \(y \in R\). If \(f ^{\prime}(0)=2\), then \(|f(-2)|\) is equal to \(.........\).JEE Mains 2023 Hard
- For some \(\theta \in\left(0, \frac{\pi}{2}\right),\) if the eccentricity of the hyperbola, \(x^{2}-y^{2} \sec ^{2} \theta=10\) is \(\sqrt{5}\) times the eccentricity of the ellipse, \(x^{2} \sec ^{2} \theta+y^{2}=5,\) then the length of the latus rectum of the ellipse isJEE Mains 2020 Hard
More PYQs from JEE Mains
- If \(\sum_{r=1}^{10} r !\left( r ^{3}+6 r ^{2}+2 r +5\right)=\alpha(11 !),\) then the value of \(\alpha\) is equal to ...... .JEE Mains 2021 Hard
- The value of
\(\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}}{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}+e^{\left(\left(6-\log _e x\right)^2+1\right)^{-1}}}\right) d x\) isJEE Mains 2025 Medium - If the coefficients of \(x\) and \(x^{2}\) in the expansion of \((1+x)^{p}(1-x)^{q}, p, q \leq 15\), are \(-3\) and \(-5\) respectively, then the coefficient of \(x ^{3}\) is equal to \(............\)JEE Mains 2022 Hard
- \(\lim _{n \rightarrow \infty}\left(1+\frac{1+\frac{1}{2}+\ldots \ldots .+\frac{1}{n}}{n^{2}}\right)^{n}=.........\)JEE Mains 2021 Hard
- Let \(\alpha, \beta, \gamma, \delta \in \mathrm{Z}\) and let \(\mathrm{A}(\alpha, \beta), \mathrm{B}(1,0), \mathrm{C}(\gamma, \delta)\) and \(D(1,2)\) be the vertices of a parallelogram \(\mathrm{ABCD}\). If \(\mathrm{AB}=\sqrt{10}\) and the points \(\mathrm{A}\) and \(\mathrm{C}\) lie on the line \(3 y=2 x+1\), then \(2(\alpha+\beta+\gamma+\delta)\) is equal toJEE Mains 2024 Hard
- Let \(f\) be a twice differentiable function such that \(f(x)=\int_{0}^{x}\tan(t-x)dt-\int_{0}^{x}f(t)\tan t\,dt\), \(x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\). Then \(f''\left(\dfrac{\pi}{6}\right)+12f'\left(-\dfrac{\pi}{6}\right)+f\left(\dfrac{\pi}{6}\right)\) is equal to ______JEE Mains 2026 Hard