JEE Mains · Maths · STD 12 - 9. differential equations
If the curve \(y=y(x)\) is the solution of the differential equation \(2\left(x^{2}+x^{5 / 4}\right) d y-y\left(x+x^{1 / 4}\right) d x=2 x^{9 / 4} d x, x > 0\) which passes through the point \(\left(1,1-\frac{4}{3} \log _{e} 2\right),\) then the value of \(y(16)\) is equal to :
- A \(4\left(\frac{31}{3}+\frac{8}{3} \log _{ e } 3\right)\)
- B \(\left(\frac{31}{3}+\frac{8}{3} \log _{ e } 3\right)\)
- C \(4\left(\frac{31}{3}-\frac{8}{3} \log _{e} 3\right)\)
- D \(\left(\frac{31}{3}-\frac{8}{3} \log _{ e } 3\right)\)
Answer & Solution
Correct Answer
(C) \(4\left(\frac{31}{3}-\frac{8}{3} \log _{e} 3\right)\)
Step-by-step Solution
Detailed explanation
\(\frac{ dy }{ dx }-\frac{ y }{2 x }=\frac{ x ^{9 / 4}}{ x ^{5 / 4}\left( x ^{3 / 4}+1\right)}\) \(IF = e ^{-\int \frac{ dx }{2 d }}= e ^{-\frac{1}{2} \ln x }=\frac{1}{ x ^{1 / 2}}\)…
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 \(S\) be the sum of all solutions (in radians) of the equation \(\sin ^{4} \theta+\cos ^{4} \theta-\sin \theta \cos \theta=0\) in \([0,4 \pi]\) Then \(\frac{8 \mathrm{~S}}{\pi}\) is equal to ...... .JEE Mains 2021 Hard
- Equation of the line of the shortest distance between the lines \(\frac{x}{1} = \frac{y}{{ - 1}} = \frac{z}{1}\) and \(\frac{{x - 1}}{0} = \frac{{y + 1}}{{ - 2}} = \frac{z}{1}\) isJEE Mains 2014 Hard
- Let \(f(x)=\int \frac{d x}{x^{\left(\frac{2}{3}\right)}+2 x^{\left(\frac{1}{2}\right)}}\) be such that \(f(0)=-26+24 \log _{ e }(2)\). If \(f (1)= a + b \log _{ e }(3)\), where \(a , b \in Z\), then \(a + b\) is equal to:JEE Mains 2026 Hard
- If the mean and the variance of the data
are \(\mu\) and 19 respectively, then the value of \(\lambda+\mu\) isClass 4-8 8-12 12-16 16-20 Frequency 3 \(\lambda\) 4 7 JEE Mains 2026 Medium - Let \(f(x)=\lim _{\theta \rightarrow 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\theta}\right)} \sin (x-1)}{1+x^{\left(\frac{2}{\theta}\right)}(x-1)}\right), x \in R\).
Consider the following two statements :
(I) \(f ( x )\) is discontinous at \(x =1\).
(II) \(f ( x )\) is continous at \(x =-1\). Then,JEE Mains 2026 Easy - The sum of all the elements in the range of \(f(x)=\text{Sgn}(\sin x)+\text{Sgn}(\cos x)+\text{Sgn}(\tan x)+\text{Sgn}(\cot x), x\ne\frac{n\pi}{2}, n\in Z\), where \(\operatorname{Sgn}(t)=\left\{\begin{array}{lll}1, & \text { if } & t>0 \\ -1 & \text { if } & t<0\end{array}\right.\), isJEE Mains 2026 Medium
More PYQs from JEE Mains
- For the function \(\mathrm{f}(\mathrm{x})=(\cos \mathrm{x})-\mathrm{x}+1, \mathrm{x} \in \mathbb{R}\), between the following two statements (\(S1\)) \(f(x)=0\) for only one value of \(x\) is \([0, \pi]\). (\(S2\)) \(\mathrm{f}(\mathrm{x})\) is decreasing in \(\left[0, \frac{\pi}{2}\right]\) and increasing in \(\left[\frac{\pi}{2}, \pi\right] .\)JEE Mains 2024 Medium
- The area of the region \(A=\left\{(x, y):|\cos x-\sin x| \leq y \leq \sin x, 0 \leq x \leq \frac{\pi}{2}\right\}\)JEE Mains 2023 Hard
- If \(P=\left[\begin{array}{ll}1 & 0 \\ 1 / 2 & 1\end{array}\right]\), then \(P^{50}\) is:JEE Mains 2021 Medium
- The natural number \(m\), for which the coefficient of \(x\) in the binomial expansion of \(\left( x ^{ m }+\frac{1}{ x ^{2}}\right)^{22}\) is \(1540,\) isJEE Mains 2020 Hard
- A coin is biased so that the head is \(3\) times as likely to occur as tail. This coin is tossed until a head or three tails occur. If \(X\) denotes the number of tosses of the coin, then the mean of \(X\) isJEE Mains 2023 Medium
- Let for a differentiable function \(f:(0, \infty) \rightarrow R\), \(f(x)-f(y) \geq \log _e\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty) \text {. }\) Then \(\sum_{\mathrm{n}=1}^{20} \mathrm{f}^{\prime}\left(\frac{1}{\mathrm{n}^2}\right)\) is equal toJEE Mains 2024 Hard