
What is the solution of the differential equation \[\cos x \cos y \dfrac{{dy}}{{dx}} = - \sin x \sin y \]?
A. \[\sin y + \cos x = c\]
B. \[\sin y - \cos x = c\]
C. \[\sin y \cos x = c\]
D. \[\sin y = c \cos x\]
Answer
162.9k+ views
Hint: Here, the first order differential equation is given. First, simplify the given equation. Then, integrate both sides of the equation with respect to the corresponding variables. After that, solve both integrals by using the standard integral rule. In the end, apply the properties of a logarithm to get the solution of the differential equation.
Formula used:
\[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\]
\[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]
Integration rule: \[\int {\dfrac{{f'\left( x \right)}}{{f\left( x \right)}}dx} = \left[ {\log f\left( x \right)} \right] + c\] , where \[c\] is an integration constant.
Complete step by step solution:
The given differential equation is \[\cos x \cos y \dfrac{{dy}}{{dx}} = - \sin x \sin y \].
Let’s find out the solution of the given differential equation.
Simplify the given equation.
\[\dfrac{{\cos y}}{{\sin y}}dy = - \dfrac{{\sin x}}{{\cos x}} dx\]
Now integrate both sides with respect to the corresponding variables.
\[\int {\dfrac{{\cos y}}{{\sin y}}dy} = \int { - \dfrac{{\sin x}}{{\cos x}} dx} \]
We know the integration rule \[\int {\dfrac{{f'\left( x \right)}}{{f\left( x \right)}}dx} = \left[ {\log f\left( x \right)} \right] + c\].
Apply this rule to solve both integrals.
\[\log\left( {\sin y} \right) = \log\left( {\cos x} \right) + \log c\]
Apply the logarithmic property of addition \[\log\left( a \right) + \log\left( b \right) = \log\left( {ab} \right)\].
We get,
\[\log\left( {\sin y} \right) = \log\left( {c \cos x} \right)\]
Equating both sides, we get
\[\sin y = c \cos x\]
Hence the correct option is D.
Note: Students often make mistakes when integrating \[ \int {- \dfrac{{\sin x}}{{\cos x}} dx}\]. They take that \[\sin x\] is the derivative of \[\cos x\]. For this reason they get \[\sin y \cos x = c\] as an answer. But the correct answer is \[\sin y = c \cos x\].
Formula used:
\[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\]
\[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]
Integration rule: \[\int {\dfrac{{f'\left( x \right)}}{{f\left( x \right)}}dx} = \left[ {\log f\left( x \right)} \right] + c\] , where \[c\] is an integration constant.
Complete step by step solution:
The given differential equation is \[\cos x \cos y \dfrac{{dy}}{{dx}} = - \sin x \sin y \].
Let’s find out the solution of the given differential equation.
Simplify the given equation.
\[\dfrac{{\cos y}}{{\sin y}}dy = - \dfrac{{\sin x}}{{\cos x}} dx\]
Now integrate both sides with respect to the corresponding variables.
\[\int {\dfrac{{\cos y}}{{\sin y}}dy} = \int { - \dfrac{{\sin x}}{{\cos x}} dx} \]
We know the integration rule \[\int {\dfrac{{f'\left( x \right)}}{{f\left( x \right)}}dx} = \left[ {\log f\left( x \right)} \right] + c\].
Apply this rule to solve both integrals.
\[\log\left( {\sin y} \right) = \log\left( {\cos x} \right) + \log c\]
Apply the logarithmic property of addition \[\log\left( a \right) + \log\left( b \right) = \log\left( {ab} \right)\].
We get,
\[\log\left( {\sin y} \right) = \log\left( {c \cos x} \right)\]
Equating both sides, we get
\[\sin y = c \cos x\]
Hence the correct option is D.
Note: Students often make mistakes when integrating \[ \int {- \dfrac{{\sin x}}{{\cos x}} dx}\]. They take that \[\sin x\] is the derivative of \[\cos x\]. For this reason they get \[\sin y \cos x = c\] as an answer. But the correct answer is \[\sin y = c \cos x\].
Recently Updated Pages
Fluid Pressure - Important Concepts and Tips for JEE

JEE Main 2023 (February 1st Shift 2) Physics Question Paper with Answer Key

Impulse Momentum Theorem Important Concepts and Tips for JEE

Graphical Methods of Vector Addition - Important Concepts for JEE

JEE Main 2022 (July 29th Shift 1) Chemistry Question Paper with Answer Key

JEE Main 2023 (February 1st Shift 1) Physics Question Paper with Answer Key

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JoSAA JEE Main & Advanced 2025 Counselling: Registration Dates, Documents, Fees, Seat Allotment & Cut‑offs

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More

Verb Forms Guide: V1, V2, V3, V4, V5 Explained

1 Billion in Rupees

Which one is a true fish A Jellyfish B Starfish C Dogfish class 11 biology CBSE
