MHT CET · Maths · Differential Equations
The solution of the differential equation \(\sin ^{-1}\left(\frac{\mathrm{dy}}{\mathrm{d} x}\right)=x+\mathrm{y}\) is
- A \(x=\tan (x+y) \cdot \sec (x+y)+c\)
- B \(x=\tan (x+y)-\sec (x+y)+c\)
- C \(x=\tan (x+y)+\sec (x+y)+c\)
- D \(x=\tan x \cdot \tan y+c\)
Answer & Solution
Correct Answer
(B) \(x=\tan (x+y)-\sec (x+y)+c\)
Step-by-step Solution
Detailed explanation
We have \(\sin ^{-1}\left(\frac{d y}{d x}\right)=x+y\)
\(
\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\sin (\mathrm{x}+\mathrm{y})
\)
Put \(\quad x+y=t \Rightarrow y=t-x \Rightarrow \frac{d y}{d x}=\frac{d t}{d x}-1\)
\(
\therefore \frac{\mathrm{dt}}{\mathrm{dx}}=1+\sin \mathrm{t} \Rightarrow \int \frac{\mathrm{dt}}{1+\sin \mathrm{t}}=\int \mathrm{dx}
\)
\(\int \frac{(1-\sin t)}{(1+\sin t)(1-\sin t)} d t=\int d x \Rightarrow x=\int \frac{1-\sin t}{1-\sin ^{2} t} d t\)
\(x=\int \frac{1-\sin t}{\cos ^{2} t} d t=\int\left(\sec ^{2} t-\sec t \tan t\right) d t\)
\(x=\tan t-\sec t+c\)
\(x=\tan (x+y)-\sec (x+y)+c\)
\(
\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\sin (\mathrm{x}+\mathrm{y})
\)
Put \(\quad x+y=t \Rightarrow y=t-x \Rightarrow \frac{d y}{d x}=\frac{d t}{d x}-1\)
\(
\therefore \frac{\mathrm{dt}}{\mathrm{dx}}=1+\sin \mathrm{t} \Rightarrow \int \frac{\mathrm{dt}}{1+\sin \mathrm{t}}=\int \mathrm{dx}
\)
\(\int \frac{(1-\sin t)}{(1+\sin t)(1-\sin t)} d t=\int d x \Rightarrow x=\int \frac{1-\sin t}{1-\sin ^{2} t} d t\)
\(x=\int \frac{1-\sin t}{\cos ^{2} t} d t=\int\left(\sec ^{2} t-\sec t \tan t\right) d t\)
\(x=\tan t-\sec t+c\)
\(x=\tan (x+y)-\sec (x+y)+c\)
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
- Scalar projection of the line segment joining the points \(\mathrm{A}(-2,0,3), \mathrm{B}(1,4,2)\) on the line whose direction ratios are \(6,-2,3\) isMHT CET 2023 Easy
- If \(\mathrm{f}(x)=\left(\frac{2^{x}-1}{1-3^{x}}\right)\), for \(\mathrm{x} \neq 0\) is continuous at \(x=0\), then \(\mathrm{f}(0)=\)MHT CET 2020 Easy
- The solution set of the constraints \(|x-y| \leqslant 1, x, y \geqslant 0\) isMHT CET 2025 Medium
- The value of \(\int_0^\pi\left|\sin x-\frac{2 x}{\pi}\right| \mathrm{d} x\) isMHT CET 2023 Medium
- The lines \(\frac{\mathrm{x}-3}{1}=\frac{\mathrm{y}-2}{1}=\frac{\mathrm{z}-5}{-\mathrm{k}}\) and \(\frac{\mathrm{x}-4}{\mathrm{k}}=\frac{\mathrm{y}-3}{1}=\frac{\mathrm{z}-3}{2}\) are coplanar, hence \(\mathrm{k}=\)MHT CET 2025 Medium
- If \(\frac{x^2}{\mathrm{a}}+\frac{2 x y}{\mathrm{~h}}+\frac{y^2}{\mathrm{~b}}=0\) represents a pair of straight lines and slope of one of the lines is twice that of the other, then \(a b: h^2\) is
[Note: The question has been modified to get the correct answer.]MHT CET 2024 Medium
More PYQs from MHT CET
- A circular current carrying coil has radius \(\mathrm{R}\). At what distance from the centre of the coil on the axis, the magnetic induction will become \(\frac{1}{8}\) th of its value at the centre of the coil?MHT CET 2020 Hard
- At absolute zero temperature, pure silicon behaves asMHT CET 2020 Easy
- For the system \(x-y+z=4,2 x+y-3 z=0\), \(x+y+z=2\), the values of \(x, y, z\) respectively are given byMHT CET 2024 Medium
- Which of the following statement is not about ionic solidMHT CET 2022 Medium
- The number of positive integral solutions of \(\tan ^{-1} x+\cos ^{-1}\left(\frac{y}{\sqrt{1+y^2}}\right)=\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)\) areMHT CET 2025 Medium
- A monochromatic ray of light is incident normally on a thin prism of refracting angle A . The ray is deviated through an angle \((1 \cdot 15)^{\circ}\) in passing through the prism. The ray reflected internally from the second face emerges from the first face making an angle of \((6 \cdot 3)^{\circ}\) with the incident ray. The refractive index of the prism isMHT CET 2025 Hard