JEE Mains · Maths · STD 12 - 9. differential equations
The solution of the differential equation \(\frac{d y}{d x}=-\left(\frac{x^2+3 y^2}{3 x^2+y^2}\right), y(1)=0\) is
- A \(\log _e|x+y|-\frac{x y}{(x+y)^2}=0\)
- B \(\log _e|x+y|+\frac{x y}{(x+y)^2}=0\)
- C \(\log _e|x+y|+\frac{2 x y}{(x+y)^2}=0\)
- D \(\log _{ e }|x+y|-\frac{2 x y}{(x+y)^2}=0\)
Answer & Solution
Correct Answer
(C) \(\log _e|x+y|+\frac{2 x y}{(x+y)^2}=0\)
Step-by-step Solution
Detailed explanation
Put \(y=v x\) \(v+x \frac{d v}{d x}=-\left(\frac{1+3 v^2}{3+v^2}\right)\) \(x \frac{d v}{d x}=-\frac{(v+1)^3}{3+v^2}\) \(\frac{\left(3+v^2\right) d v}{(v+1)^3}+\frac{d x}{x}=0\) \(\int \frac{4 d v}{(v+1)^3}+\int \frac{d v}{v+1}-\int \frac{2 d v}{(v+1)^2}+\int \frac{d x}{x}=0\)…
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
- Let \(a_{1}, a_{2}, a_{3}, \ldots\) be an \(A.P.\) If \(\sum_{r=1}^{\infty} \frac{a_{r}}{2^{r}}=4\), then \(4 a_{2}\) is equal toJEE Mains 2022 Hard
- All the pairs \((x, y)\) that satisfy the inequality \({2^{\sqrt {{{\sin }^2}{\kern 1pt} x - 2\sin {\kern 1pt} x + 5} }}.\frac{1}{{{4^{{{\sin }^2}\,y}}}} \leq 1\) also Satisfy the equationJEE Mains 2019 Hard
- A survey shows that \(73 \%\) of the persons working in an office like coffee, whereas \(65 \%\) like tea. If \(x\) denotes the percentage of them, who like both coffee and tea, then \(x\) cannot beJEE Mains 2020 Medium
- If the shortest distance between the line joining the points \((1, 2, 3)\) and \((2,3,4)\), and the line \(\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-2}{0}\) is \(\alpha\), then \(28 \alpha^2\) is equal to \(........\).JEE Mains 2023 Hard
- If the line \(y =4+ kx , k >0\), is the tangent to the parabola \(y = x - x ^{2}\) at the point \(P\) and \(V\) is the vertex of the parabola, then the slope of the line through \(P\) and \(V\) isJEE Mains 2022 Hard
- \(ABC\) is a triangular park with \(AB = AC = 100\) \(metres\). A vertical tower is situated at the mid-point of \(BC\). If the angles of elevation of the top of the tower at \(A\) and \(B\) are \({\cot ^{ - 1}}\left( {3\sqrt 2 } \right)\) and \(\cos e{c^{ - 1}}\left( {2\sqrt 2 } \right)\) respectively, then the height of the tower (in metres) isJEE Mains 2019 Hard
More PYQs from JEE Mains
- The foot of the perpendicular drawn from the origin, on the line, \(3x + y = \lambda \,\left( {\lambda \ne 0} \right)\) is \(P\). If the line meets \(x-\) axis at \(A\) and \(y-\) axis at \(B\), then the ratio \(BP : PA\) isJEE Mains 2018 Hard
- A committee of \(4\) persons is to be formed from \(2\) ladies, \(2\) old men and \(4\) young men such that it includes at least \(1\) lady, at least \(1\) old man and at most \(2\) young men. Then the total number of ways in which this committee can be formed isJEE Mains 2013 Hard
- Let \(A\) be any \(3 \times 3\) invertible matrix. Then which one of the following is not always true ?JEE Mains 2017 Hard
- Let \(f\) be a differentiable function \(R\) to \(R\) such that \(\left| {f\,(x)\, - \,f(y)} \right|\, \le \,2\,{\left| {x - y} \right|^{\frac{3}{2}}},\) for all \(x,y\,\in R .\) If \(f\,(0)=1\) then \(\int\limits_0^1 {{f^2}\,(x)\,dx} \) is equal toJEE Mains 2019 Hard
- If the sum of the squares of the reciprocals of the roots \(\alpha\) and \(\beta\) of the equation \(3 x^{2}+\lambda x-1=0\) is 15 , then \(6\left(\alpha^{3}+\beta^{3}\right)^{2}\) is equal toJEE Mains 2022 Hard
- Let \(\mathrm{Y}=\mathrm{Y}(\mathrm{X})\) be a curve lying in the first quadrant such that the area enclosed by the line \(Y-y=Y^{\prime}(x)(X-x)\) and the co-ordinate axes, where \((\mathrm{x}, \mathrm{y})\) is any point on the curve, is always \(\frac{-y^2}{2 Y^{\prime}(x)}+1, Y^{\prime}(x) \neq 0\). If \(Y(1)=1\), then \(12 Y(2)\) equalsJEE Mains 2024 Hard