GATE Physics Exam > GATE Physics Questions > The solution of the differential equation (1 ...
Start Learning for Free
The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0
(a) x = y ^ 3 + cy
(b) x ^ 2 = y ^ 2 + cy
(c) x = (y ^ 2)/2 + c
(d) x = y ^ 4 + cy?
(a) x = y ^ 3 + cy
(b) x ^ 2 = y ^ 2 + cy
(c) x = (y ^ 2)/2 + c
(d) x = y ^ 4 + cy?
Most Upvoted Answer
The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^...
Understanding the Differential Equation
The given differential equation is
(1 + y^2) * dx + (x - tan^(-1)(y)) * dy = 0.
This equation is separable and can be rearranged for integration.
Separation of Variables
1. Rearranging the equation, we get:
- dy/dx = - (1 + y^2) / (x - tan^(-1)(y)).
2. This form allows us to separate the variables y and x, facilitating integration.
Integrating Both Sides
1. Integrate the left side with respect to y and the right side with respect to x:
- Integral of (1 + y^2) dy relates to y and gives us a function of y.
- Integral of (x - tan^(-1)(y)) dx relates to x.
Identifying Solutions
After integration, we need to rearrange the equation to identify the form of the solution.
- The potential solutions given are:
- (a) x = y^3 + c
- (b) x^2 = y^2 + c
- (c) x = (y^2)/2 + c
- (d) x = y^4 + c
By analyzing the integrals and considering the structure of the solution forms, we can determine which matches the integrated expression.
Final Solution Selection
The correct solution from the options provided is:
- (b) x^2 = y^2 + c.
This fits with the derived relationship from the integration process, confirming it as a valid solution to the original differential equation.
The given differential equation is
(1 + y^2) * dx + (x - tan^(-1)(y)) * dy = 0.
This equation is separable and can be rearranged for integration.
Separation of Variables
1. Rearranging the equation, we get:
- dy/dx = - (1 + y^2) / (x - tan^(-1)(y)).
2. This form allows us to separate the variables y and x, facilitating integration.
Integrating Both Sides
1. Integrate the left side with respect to y and the right side with respect to x:
- Integral of (1 + y^2) dy relates to y and gives us a function of y.
- Integral of (x - tan^(-1)(y)) dx relates to x.
Identifying Solutions
After integration, we need to rearrange the equation to identify the form of the solution.
- The potential solutions given are:
- (a) x = y^3 + c
- (b) x^2 = y^2 + c
- (c) x = (y^2)/2 + c
- (d) x = y^4 + c
By analyzing the integrals and considering the structure of the solution forms, we can determine which matches the integrated expression.
Final Solution Selection
The correct solution from the options provided is:
- (b) x^2 = y^2 + c.
This fits with the derived relationship from the integration process, confirming it as a valid solution to the original differential equation.
|
Explore Courses for GATE Physics exam
|
|
Similar GATE Physics Doubts
Top Courses for GATE PhysicsView all
The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy?
Question Description
The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy?.
The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy?.
Solutions for The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy? in English & in Hindi are available as part of our courses for GATE Physics.
Download more important topics, notes, lectures and mock test series for GATE Physics Exam by signing up for free.
Here you can find the meaning of The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy? defined & explained in the simplest way possible. Besides giving the explanation of
The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy?, a detailed solution for The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy? has been provided alongside types of The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy? theory, EduRev gives you an
ample number of questions to practice The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1(y))) * dy = 0(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy? tests, examples and also practice GATE Physics tests.
|
|
Explore Courses for GATE Physics exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup