MHT CET · Maths · Differential Equations
The particular solution of the differential equation \(\left(y+x \cdot \frac{d y}{d x}\right) \cdot \sin x y=\cos x\)
at \(x=0\) is
- A \(\sin x+\cos x y=1\)
- B \(\cos x-\sin x y=1\)
- C \(\sin x-\cos x y=1\)
- D \(\cos x+\sin x y=1\)
Answer & Solution
Correct Answer
(A) \(\sin x+\cos x y=1\)
Step-by-step Solution
Detailed explanation
We have \(\left(y+x \frac{d y}{d x}\right) \sin x y=\cos x\)
Put \(x y=u \Rightarrow x \frac{d y}{d x}+y=\frac{d u}{d x}\)
\(\therefore\left(\frac{\mathrm{du}}{\mathrm{dx}}\right) \sin \mathrm{u}=\cos \mathrm{x}\)
\(\therefore \int \sin u d u=\int \cos x d x \Rightarrow-\cos u=\sin x+c \Rightarrow\) \(-\cos x y=\sin x+c\)
When \(x=0\), we get
\(-\cos 0=0+c \Rightarrow c=-1\)
\(\therefore-\cos x y=\sin x-1 \Rightarrow \sin x+\cos x y=1\)
Put \(x y=u \Rightarrow x \frac{d y}{d x}+y=\frac{d u}{d x}\)
\(\therefore\left(\frac{\mathrm{du}}{\mathrm{dx}}\right) \sin \mathrm{u}=\cos \mathrm{x}\)
\(\therefore \int \sin u d u=\int \cos x d x \Rightarrow-\cos u=\sin x+c \Rightarrow\) \(-\cos x y=\sin x+c\)
When \(x=0\), we get
\(-\cos 0=0+c \Rightarrow c=-1\)
\(\therefore-\cos x y=\sin x-1 \Rightarrow \sin x+\cos x y=1\)
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
- If \(e_{1}\) is the eccentricity of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b\) and \(e_{2}\) is the eccentricity
of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), then \(e_{1}^{2}+e_{2}^{2}=\)MHT CET 2020 Easy - 20 meters wire is available to fence a flower bed in the form of a circular sector. If the flower bed should have the greatest possible surface area, then the radius of the circle isMHT CET 2020 Easy
- If \(A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & -\cos \theta\end{array}\right]\), then \(A^{-1}=\)MHT CET 2020 Easy
- If \(\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi\), then the value of \(x^{2025}+x^{2026}+x^{2027}\) isMHT CET 2023 Easy
- The base of an equilateral triangle is represented by the equation \(2 x-y-1=0\) and its vertex is \((1,2)\), then the length (in units) of the side of the triangle isMHT CET 2023 Medium
- If the angle between the line \(x=\frac{\mathrm{y}-1}{2}=\frac{\mathrm{z}-3}{\lambda}\) and the plane \(x+2 y+3 z=4\) is \(\cos ^{-1} \sqrt{\frac{5}{14}}\), then the value of \(\lambda\) isMHT CET 2025 Medium
More PYQs from MHT CET
- In angiosperms, the generative cell inside the pollen grain divides to form __________ .MHT CET 2023 Medium
- "A given compound always contains the same proportion of elements" is a statement of -MHT CET 2024 Easy
- A differential equation for the temperature ' \(\mathrm{T}\) ' of a hot body as a function of time, when it is placed in a both which is held at a constant temperature of \(32^{\circ} \mathrm{F}\), is given by (where \(\mathrm{k}\) is a constant of proportionality)MHT CET 2021 Hard
- Water rises up to height ' \(x\) ' in a capillary tube immersed vertically in water. When the whole arrangement is taken to a depth ' d ' in a mine, the water level rises up to height Y . If R is the radius of the earth then the ratio \(\mathrm{Y}: \mathrm{x}\) isMHT CET 2025 Medium
- A string of length ' \(L\) ' fixed at one end carries a body of mass ' \(m\) ' at the other end. The mass is revolved in a circle in the horizontal plane about a vertical axis passing through the fixed end of the string. The string makes angle ' \(\theta\) ' with the vertical. The angular frequency of the body is ' \(\omega\) '. The tension in the string isMHT CET 2023 Medium
- A block of mass ‘m’ moving on a frictionless surface at speed ‘v’ collides elastically with a block of same mass, initially at rest. Now the first block moves at an angle ‘θ’ with its initial direction and has speed ‘’ . The speed of the second block after collision isMHT CET 2019 Hard