MHT CET · Maths · Differential Equations
The general solution of the differential equation \(\frac{d y}{d x}=\frac{3 e^{2 x}+3 e^{4 x}}{. e^x+e^{-x}}\) is
- A \(y=\mathrm{e}^{-3 x}+\mathrm{c}\), where c is a constant of integration.
- B \(y=\mathrm{e}^x+\mathrm{c}\), where c is a constant of integration.
- C \(y=\mathrm{e}^{3 x}+\mathrm{c}\), where c is a constant of integration.
- D \(y=\mathrm{e}^{-x}+\mathrm{c}\), where c is a constant of integration.
Answer & Solution
Correct Answer
(C) \(y=\mathrm{e}^{3 x}+\mathrm{c}\), where c is a constant of integration.
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
& \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3 \mathrm{e}^{2 x}+3 \mathrm{e}^{4 x}}{\mathrm{e}^x+\mathrm{e}^{-x}} \\
& \Rightarrow \frac{\mathrm{~d} y}{\mathrm{~d} x}=\frac{3 \mathrm{e}^{2 x}\left(1+\mathrm{e}^{2 x}\right)}{\left(\frac{\mathrm{e}^{2 x}+1}{\mathrm{e}^x}\right)} \\
& \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=3 \mathrm{e}^{2 x} \cdot \mathrm{e}^x \\
& \Rightarrow \mathrm{~d} y=3 \mathrm{e}^{3 x} \mathrm{~d} x
\end{aligned}\)
Integrating on both sides, we get
\(y=\mathrm{e}^{3 x}+\mathrm{c}\)
& \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3 \mathrm{e}^{2 x}+3 \mathrm{e}^{4 x}}{\mathrm{e}^x+\mathrm{e}^{-x}} \\
& \Rightarrow \frac{\mathrm{~d} y}{\mathrm{~d} x}=\frac{3 \mathrm{e}^{2 x}\left(1+\mathrm{e}^{2 x}\right)}{\left(\frac{\mathrm{e}^{2 x}+1}{\mathrm{e}^x}\right)} \\
& \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=3 \mathrm{e}^{2 x} \cdot \mathrm{e}^x \\
& \Rightarrow \mathrm{~d} y=3 \mathrm{e}^{3 x} \mathrm{~d} x
\end{aligned}\)
Integrating on both sides, we get
\(y=\mathrm{e}^{3 x}+\mathrm{c}\)
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 general solution of differential equation isMHT CET 2018 Hard
- If \(\mathrm{A}=\left[\begin{array}{rr}1 & 2 \\ -1 & 4\end{array}\right]\) and \(\mathrm{A}^{-1}=\alpha \mathrm{I}+\beta \mathrm{A}\)
\(\alpha, \beta \in \mathrm{R}\) where I is the identity matrix of order 2 , then \(4(\alpha+\beta)=\)MHT CET 2025 Medium - Local maximum and local minimum values respectively of the function \(f(x)=(x-1)(x+2)^2\) areMHT CET 2022 Easy
- If the lines \(\frac{2 x-4}{\lambda}=\frac{y-1}{2}=\frac{z-3}{1}\) and \(\frac{x-1}{1}=\frac{3 y-1}{\lambda}=\frac{z-2}{1}\) are perpendicular to each other, then \(\lambda=\)MHT CET 2021 Easy
- The co-efficient of \(x^{6}\) in the series of \(e^{2 x}\) isMHT CET 2020 Medium
- The equation of line passing through and perpendicular to the lines and isMHT CET 2018 Easy
More PYQs from MHT CET
- The materials suitable for making electromagnets should haveMHT CET 2023 Easy
- \(\int \frac{\mathrm{d} x}{(x+a)^{\frac{9}{7}}(x-b)^{5 / 7}}=\)MHT CET 2025 Medium
- Calculate the work done in the following reaction at \(300 \mathrm{~K}\) and at constant pressure.
\(\left(\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\right) 4 \mathrm{HCl}_{(\mathrm{g})}+\) \(\mathrm{O}_{2(\mathrm{~g})} \rightarrow 2 \mathrm{Cl}_{2(\mathrm{~g})}+2 \mathrm{H}_2 \mathrm{O}_{(\mathrm{g})}\)MHT CET 2023 Medium - In an organ pipe closed at one end, the sum of the frequencies of first three overtones is 3930 Hz . The frequency of the fundamental mode of organ pipe isMHT CET 2025 Medium
- Find a polynomial \(f(x)\) of degree 2 where \(f(0)=8, f(1)=12, f(2)=18\)MHT CET 2009 Easy
- If \(x=\sec \theta, y=\tan \theta\), then the value of \(\frac{d^{2} y}{d x^{2}}\) at
\(\theta=\frac{\pi}{4}\) isMHT CET 2010 Easy