JEE Mains · Maths · STD 12 - 9. differential equations
Let \(\alpha x=\exp \left(x^\beta y^\gamma\right)\) be the solution of the differential equation \(2 x^2 y d y-\left(1-x y^2\right) d x=0\), \(x >0, y (2)=\sqrt{\log _e 2}\). Then \(\alpha+\beta-\gamma\) equals :
- A \(1\)
- B \(-1\)
- C \(0\)
- D \(3\)
Answer & Solution
Correct Answer
(A) \(1\)
Step-by-step Solution
Detailed explanation
\(\alpha x = e ^{ x ^{\cdot} \cdot y ^7}\) \(2 x ^2 y \frac{ dy }{ dx }=1- x \cdot y ^2 \quad y ^2= t\) \(x ^2 \frac{ dt }{ dx }=1- xt\) \(\frac{ dt }{ dx }+\frac{ t }{ x }=\frac{1}{ x ^2} \quad \text { I.F. }= e ^{\ell nx }= x\) \(t ( x )=\int \frac{1}{ x ^2} \cdot x dx\)…
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 range of \(a \in R\) for which the function \( f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)\) \(x \neq 2 n \pi, n \in N ,\) has critical points, isJEE Mains 2021 Hard
- The area (in sq. units) of the region \(A=\{(x, y)\) \(\left.:|x|+|y| \leq 1,2 y^{2} \geq|x|\right\}\) isJEE Mains 2020 Hard
- The system of equations \(kx + y + z =1\) \(x + ky + z = k\) and \(x + y + zk = k ^{2}\) has no solution if \(k\) is equal toJEE Mains 2021 Medium
- If \(y=y(x)\) is the solution of the equaiton \(e ^{\sin y} \cos y \frac{ dy }{ dx }+ e ^{\sin y} \cos x =\cos x , y (0)=0\) then \(1+ y \left(\frac{\pi}{6}\right)+\frac{\sqrt{3}}{2} y \left(\frac{\pi}{3}\right)+\frac{1}{\sqrt{2}} y \left(\frac{\pi}{4}\right)\) is equal toJEE Mains 2021 Hard
- The total number of six digit numbers, formed using the digits \(4,5,9\) only and divisible by \(6\) , is \(.........\).JEE Mains 2023 Hard
- Let \(\overrightarrow{ a }=\alpha \hat{ i }+2 \hat{ j }-\hat{ k }\) and \(\overrightarrow{ b }=-2 \hat{ i }+\alpha \hat{ j }+\hat{ k }\), where \(\alpha \in R\). If the area of the parallelogram whose adjacent sides are represented by the vectors \(\vec{a}\) and \(\vec{b}\) is \(\sqrt{15\left(\alpha^{2}+4\right)}\), then the value of \(2|\vec{a}|^{2}+(\vec{a} \cdot \vec{b})|\vec{b}|^{2}\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- Let \(a \neq b\) be two non-zero real numbers.Then the number of elements in the set \(X =\left\{ z \in C : \operatorname{Re}\left(a z^2+ bz \right)= a \text { and }\operatorname{Re}\left(b z^2+ az \right)= b \right\}\) is equal toJEE Mains 2023 Hard
- If the mean and the standard deviation of the data \(3,5,7, a, b\) are \(5\) and \(2\) respectively, then \(a\) and \(b\) are the roots of the equationJEE Mains 2020 Hard
- A spherical iron ball of radius \(10\,cm\) is coated with a layer of ice of uniform thickness that melts at a rate of \(50\,cm^3/min.\) When the thickness of the ice is \(5\,cm,\) then the rate at which the thickness (in \(cm/min\) ) of ice decreases isJEE Mains 2020 Hard
- Let \(g ( x )=\int_{0}^{ x } f( t ) dt ,\) where \(f\) is continuous function in \([0,3]\) such that \(\frac{1}{3} \leq f(t) \leq 1\) for all \(t \in[0,1]\) and \(0 \leq f( t ) \leq \frac{1}{2}\) for all \(t \in(1,3]\) The largest possible interval in which \(g (3)\) lies is :JEE Mains 2021 Hard
- Let \(f: R-\left\{\frac{\alpha}{6}\right\} \rightarrow R\) be defined by \(f(x)=\frac{5 x+3}{6 x-\alpha} .\) Then the value of \(\alpha\) for which \((fof)(x)=x\), for all \(x \in R-\left\{\frac{\alpha}{6}\right\}\), is:JEE Mains 2021 Medium
- Let \(\mu\) be the mean and \(\sigma\) be the standard deviation of the distribution
where \(\sum f_i=62\). if \([x]\) denotes the greatest integer \(\leq x\), then \(\left[\mu^2+\sigma^2\right]\) is equal \(.........\).\(X_i\) \(0\) \(1\) \(2\) \(3\) \(4\) \(5\) \(f_i\) \(k+2\) \(2k\) \(K^{2}-1\) \(K^{2}-1\) \(K^{2}-1\) \(k-3\) JEE Mains 2023 Hard