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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 can be rearranged to:
dx / (x - tan^-1(y)) = -dy / (1 + y^2)
This format suggests that we can separate variables for integration.
Identifying the Integrating Factors
To solve this equation, we can observe the structure of the terms involved. The left-hand side involves a function of x and the right-hand side involves a function of y.
We need to integrate both sides:
- The integral of dx / (x - tan^-1(y)) will depend on the function x, while
- The integral of -dy / (1 + y^2) will yield -tan^-1(y).
Finding the General Solution
After integrating both sides, we will find a relationship between x and y.
1. When we integrate the left side, we get:
- ln|x - tan^-1(y)| = function of y
2. The integral of -dy / (1 + y^2) will give:
- -tan^-1(y) + C
Combining these results leads to a solution that can be expressed in several forms.
Evaluating the Given Options
Now, let’s evaluate the given options:
- (a) x = y^3 + C
- (b) x^2 = y^2 + C
- (c) x = (y^2)/2 + C
- (d) x = y^4 + C
After integration and simplification, the solution that best represents the relationship derived from the differential equation is:
Final Answer
The correct solution is:
(b) x^2 = y^2 + C
This form satisfies the conditions from the differential equation and represents the relationship between x and y effectively.
The given differential equation is:
(1 + y^2) * dx + (x - tan^-1(y)) * dy = 0
This can be rearranged to:
dx / (x - tan^-1(y)) = -dy / (1 + y^2)
This format suggests that we can separate variables for integration.
Identifying the Integrating Factors
To solve this equation, we can observe the structure of the terms involved. The left-hand side involves a function of x and the right-hand side involves a function of y.
We need to integrate both sides:
- The integral of dx / (x - tan^-1(y)) will depend on the function x, while
- The integral of -dy / (1 + y^2) will yield -tan^-1(y).
Finding the General Solution
After integrating both sides, we will find a relationship between x and y.
1. When we integrate the left side, we get:
- ln|x - tan^-1(y)| = function of y
2. The integral of -dy / (1 + y^2) will give:
- -tan^-1(y) + C
Combining these results leads to a solution that can be expressed in several forms.
Evaluating the Given Options
Now, let’s evaluate the given options:
- (a) x = y^3 + C
- (b) x^2 = y^2 + C
- (c) x = (y^2)/2 + C
- (d) x = y^4 + C
After integration and simplification, the solution that best represents the relationship derived from the differential equation is:
Final Answer
The correct solution is:
(b) x^2 = y^2 + C
This form satisfies the conditions from the differential equation and represents the relationship between x and y effectively.
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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?
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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.
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