JEE Mains · Maths · STD 12 - 9. differential equations
Let \(y=y(x)\) be the solution of the differential equation \(\left(2 x \log _e x\right) \frac{d y}{d x}+2 y=\frac{3}{x} \log _e x, x>0\) and \(y\left(e^{-1}\right)=0\). Then, \(y(e)\) is equal to
- A \(-\frac{3}{2 \mathrm{e}}\)
- B \(-\frac{2}{3 e}\)
- C \(-\frac{3}{\mathrm{e}}\)
- D \(-\frac{2}{\mathrm{e}}\)
Answer & Solution
Correct Answer
(C) \(-\frac{3}{\mathrm{e}}\)
Step-by-step Solution
Detailed explanation
\( \frac{\mathrm{dy}}{\mathrm{dx}}+\frac{\mathrm{y}}{\mathrm{x} \ell \operatorname{nn} x}=\frac{3}{2 \mathrm{x}^2} \) \( \therefore \text { I.F. }=\mathrm{e}^{\int \frac{1}{\mathrm{x} \ln \mathrm{x}} \mathrm{dx}}=\mathrm{e}^{\ln (\ln (\mathrm{x}))}=\ln \mathrm{x} \)…
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 integral \(\int_{\pi /6}^{\pi /4} {\frac{{dx}}{{\sin \,2x\,\left( {{{\tan }^5}\,x + {{\cot }^5}\,x} \right)}}} \) equalsJEE Mains 2019 Hard
- The tangent and normal to the ellipse \(3x^2 + 5y^2 = 32\) at the point \(P(2, 2)\) meet the \(x-\) axis at \(Q\) and \(R,\) respectively. Then the area(in sq. units) of the triangle \(PQR\) isJEE Mains 2019 Hard
- Let \(\mathrm{A}=\{1,2,3,4\}\) and \(\mathrm{B}=\{1,4,9,16\}\). Then the number of many-one functions \(f: \mathrm{A} \rightarrow \mathrm{B}\) such that \(1 \in f(\mathrm{~A})\) is equal to :JEE Mains 2025 Hard
- Let the coefficients of \(x ^{-1}\) and \(x ^{-3}\) in the expansion of \(\left(2 x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)^{15}, x>0\), be \(m\) and \(n\) respectively. If \(r\) is a positive integer such \(m n^{2}={ }^{15} C _{ r } .2^{ r }\), then the value of \(r\) is equal toJEE Mains 2022 Medium
- Let \(\mathrm{A}\) be a \(3 \times 3\) real matrix. If \(\operatorname{det}(2 \operatorname{Adj}(2 \operatorname{Adj}(\operatorname{Adj}(2 \mathrm{~A}))))=2^{41}\), then the value of \(\operatorname{det}\left(A^{2}\right)\) equal ..... .JEE Mains 2021 Hard
- If a curve the \(y = f(x)\) passes through point \((1, -1)\) and satisfies the differential equation \(y\left( {1 + xy} \right)dx = xdy\) then \(f\left( { - \frac{1}{2}} \right) = \) . . . . .JEE Mains 2016 Hard
More PYQs from JEE Mains
- If the foot of the perpendicular drawn from \((1,9\), 7) to the line passing through the point \((3,2,1)\) and parallel to the planes \(x+2 y+z=0\) and \(3 y-z=3\) is \((\alpha, \beta, \gamma)\), then \(\alpha+\beta+\gamma\) is equal toJEE Mains 2023 Hard
- Let \(f: R \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{ccc}\frac{a-b \cos 2 x}{x^2} & ; & x<0 \\ x^2+c x+2 & ; & 0 \leq x \leq 1 \\ 2 x+1 & ; & x>1\end{array}\right.\) If \(f\) is continuous everywhere in \(R\) and \(\mathrm{m}\) is the number of points where \(f\) is \(NOT\) differential then \(\mathrm{m}+\mathrm{a}+\mathrm{b}+\mathrm{c}\) equals :JEE Mains 2024 Hard
- Let the mid points of the sides of a triangle \(ABC\) be \(\left(\dfrac{5}{2}, 7\right)\), \(\left(\dfrac{5}{2}, 3\right)\) and \((4, 5)\). If its incentre is \((h, k)\), then \(3h + k\) is equal to :JEE Mains 2026 Hard
- The sum of all the local minimum values of the twice differentiable function \(\mathrm{F}: \mathrm{R} \rightarrow \mathrm{R}\) defined by \(f(x)=x^{3}-3 x^{2}-\frac{3 f^{\prime \prime}(2)}{2} x+f^{\prime \prime}(1)\) is:JEE Mains 2021 Hard
- Let O be the vertex of the parabola \(y^2=4x\) and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C, then the length of its latus rectum is:JEE Mains 2026 Hard
- If \(f(x)\) is a non-zero polynomial of degree four, having local extreme points at \(x = -1, 0, 1;\) then the set \(S = \{x \in R; f(x) = f(0)\}\) contains exactlyJEE Mains 2019 Hard