MHT CET · Maths · Differential Equations
General solution of the differential equation \(\left(y^3+y\right)\left(x^2+1\right) d y=\left(x y^4+2 y^2 x\right) d x\) is (where \(C\) is a constant of integration.)
- A \(\mathrm{y}^2\left(\mathrm{y}^2+1\right)=\mathrm{C}\left(\mathrm{x}^2+1\right)^2\)
- B \(\mathrm{y}^2\left(\mathrm{y}^2+2\right)=\mathrm{C}\left(\mathrm{x}^2+1\right)\)
- C \(\mathrm{y}^2\left(\mathrm{y}^2+2\right)=\mathrm{C}\left(\mathrm{x}^2+1\right)^2\)
- D \(y^2\left(y^2+1\right)=C\left(x^2+2\right)^2\)
Answer & Solution
Correct Answer
(C) \(\mathrm{y}^2\left(\mathrm{y}^2+2\right)=\mathrm{C}\left(\mathrm{x}^2+1\right)^2\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \left(y^3+y\right)\left(x^2+1\right) d y=\left(x y^4+2 y^2 x\right) d x \\ & \Rightarrow \int \frac{y^3+y}{y^4+2 y^2} d y=\int \frac{x}{x^2+1} d x \\ & \Rightarrow \frac{1}{4} \log _e\left(y^4+2 y^2\right)=\frac{1}{2} \log _e\left(x^2+1\right)+\log C^1 \\ & \Rightarrow \log _e\left(y^4+2 y^2\right)=2 \log _e\left(C^1\right)^2\left(x^2+1\right) \\ & \Rightarrow y^4+2 y^2=\left(c^1\right)^4\left(x^2+1\right)^2 \\ & \Rightarrow y^2\left(y^2+2\right)=C\left(x^2+1\right)^2\end{aligned}\)
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 integral values of k for which the equation \(7 \cos x+5 \sin x=2 \mathrm{k}+1\) has a solution, isMHT CET 2024 Hard
- \(\int \frac{d x}{32-2 x^2}=A \log (4-x)+B \log (4+x)+c\), then the value of \(\mathrm{A}\) and \(\mathrm{B}\) are respectively (where \(\mathrm{c}\) is a constant of integration)MHT CET 2021 Medium
- The common principal solution of the equations \(\sin \theta=-\frac{1}{2}\) and \(\tan \theta=\frac{1}{\sqrt{3}}\) isMHT CET 2025 Easy
- If \(\mathrm{f}^{\prime}(\mathrm{x})=\tan ^{-1}(\sec \mathrm{x}+ \tan \mathrm{x}), \frac{-\pi}{2}<\mathrm{x}<\frac{\pi}{2}\) and \(\mathrm{f}(0)=0\), then \(\mathrm{f}(1)=\)MHT CET 2022 Medium
- If the radius of a circle increases at the rate of \(7 \mathrm{~cm} / \mathrm{sec}\), then the rate of increase of its area after 10 minutes isMHT CET 2020 Easy
- If surrounding air is kept at \(20^{\circ} \mathrm{C}\) and body cools from \(80^{\circ} \mathrm{C}\) to \(70^{\circ} \mathrm{C}\) in 5 minutes, then the temperature of the body after 15 minute will beMHT CET 2022 Hard
More PYQs from MHT CET
- A ball of mass 'm' is attached to the free end of an inextensible string of length ' \(\ell\) '. Let 'T' be the tension in the string. The ball is moving in horizontal circular path about the vertical axis. The angular velocity of the ball at any particular instant will beMHT CET 2020 Easy
- \(\int_{-1}^{3}\left[\tan ^{-1}\left(\frac{x}{x^{2}+1}\right)+\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)\right] d x=\)MHT CET 2020 Easy
- A streamline flow of a liquid of density ' \(\rho\) ' is passing through a horizontal pipe of cross-sectional area \(A_1\) and \(A_2\) at two ends. If the pressure of liquid is ' P ' at a point where flow speed is ' \(v\) ', then pressure at another point where the flow of speed becomes 3 v isMHT CET 2024 Medium
- With reference to the principal values, if \(\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3 \pi}{2}\), then \(x^{100}+y^{100}+z^{100}=\)MHT CET 2022 Easy
- In an open end organ pipe of length ' \(L\) ', if the velocity of sound is ' \(V\) ', then the fundamental frequency will be (Neglect end correction)MHT CET 2025 Easy
- Let \(\bar{a}, \bar{b}\) and \(\bar{c}\) be three unit vectors such that \(\bar{a} \times(\bar{b} \times \bar{c})=\frac{\sqrt{3}}{2}(\bar{b}+\bar{c})\).
If \(\bar{b}\) is not parallel to \(\bar{c}\), then the angle between \(\bar{a}\) and \(\bar{b}\) isMHT CET 2024 Medium