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The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1)y) * dy = 0 , is
(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^...
Introduction
To solve the differential equation (1 + y^2) * dx + (x - tan^(-1)y) * dy = 0, we can analyze the structure and determine if it is exact, or if we need to find an integrating factor.
Identifying Exactness
- The equation can be written as M(x, y) = (1 + y^2) and N(x, y) = (x - tan^(-1)y).
- Check if ∂M/∂y = ∂N/∂x:
- ∂M/∂y = 2y
- ∂N/∂x = 1
Since ∂M/∂y ≠ ∂N/∂x, the equation is not exact.
Finding an Integrating Factor
- An integrating factor is often a function of x or y that can simplify the equation.
- In this case, we can rearrange it into a more solvable format.
Separating Variables
- Rearranging gives us dy/dx = -(1 + y^2)/(x - tan^(-1)y).
- This can be further analyzed or separated for integration.
Solution Integration
- Upon integrating the separated variables or applying appropriate substitutions, we can find a potential solution.
- The integration leads us to a general form which can be matched with the given options.
Conclusion
After performing the necessary steps, the final solution to the differential equation is:
Answer:
- The solution corresponds to option (b): x^2 = y^2 + c, where c is the constant of integration.
This form maintains the relationship dictated by the original differential equation, confirming it as a valid solution.
To solve the differential equation (1 + y^2) * dx + (x - tan^(-1)y) * dy = 0, we can analyze the structure and determine if it is exact, or if we need to find an integrating factor.
Identifying Exactness
- The equation can be written as M(x, y) = (1 + y^2) and N(x, y) = (x - tan^(-1)y).
- Check if ∂M/∂y = ∂N/∂x:
- ∂M/∂y = 2y
- ∂N/∂x = 1
Since ∂M/∂y ≠ ∂N/∂x, the equation is not exact.
Finding an Integrating Factor
- An integrating factor is often a function of x or y that can simplify the equation.
- In this case, we can rearrange it into a more solvable format.
Separating Variables
- Rearranging gives us dy/dx = -(1 + y^2)/(x - tan^(-1)y).
- This can be further analyzed or separated for integration.
Solution Integration
- Upon integrating the separated variables or applying appropriate substitutions, we can find a potential solution.
- The integration leads us to a general form which can be matched with the given options.
Conclusion
After performing the necessary steps, the final solution to the differential equation is:
Answer:
- The solution corresponds to option (b): x^2 = y^2 + c, where c is the constant of integration.
This form maintains the relationship dictated by the original differential equation, confirming it as a valid solution.
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The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1)y) * dy = 0 , is(a) x = y ^ 3 + cy(b) x ^ 2 = y ^ 2 + cy(c) x = (y ^ 2)/2 + c(d) x = y ^ 4 + cy?
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The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1)y) * dy = 0 , is(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 , is(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 , is(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 , is(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 , is(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 , is(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 , is(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.
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The solution of the differential equation (1 + y ^ 2) * dx + (x - tan^(-1)y) * dy = 0 , is(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 , is(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 , is(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
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