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The solution of the differential equatio (x ^ 3 * y ^ 3 + 1) * dx + x ^ 4 * y ^ 2 * dy = 0 , is
(a) ((xy) ^ 3)/3 + ln(x) = c
(b) ((xy) ^ 2)/2 + ln(x) = c
(c) ((xy) ^ 4)/4 + ln(x) = c
(d) (xy)/4 + ln(x) = c?
(a) ((xy) ^ 3)/3 + ln(x) = c
(b) ((xy) ^ 2)/2 + ln(x) = c
(c) ((xy) ^ 4)/4 + ln(x) = c
(d) (xy)/4 + ln(x) = c?
Most Upvoted Answer
The solution of the differential equatio (x ^ 3 * y ^ 3 + 1) * dx + x ...
Understanding the Differential Equation
The differential equation given is:
(x^3 * y^3 + 1) * dx + x^4 * y^2 * dy = 0.
To find the solution, we can rearrange it into the standard form:
dx/(x^4 * y^2) + dy/(x^3 * y^3 + 1) = 0.
This indicates a separable equation, allowing us to integrate both sides.
Integrating the Terms
1. Integration of dx term:
We integrate dx/(x^4 * y^2). This will involve y being treated as a constant during integration.
2. Integration of dy term:
For dy/(x^3 * y^3 + 1), we focus on x being a constant during integration.
After performing the integration, we find the terms relate to the product xy.
Finding the General Solution
Combining both integrations leads to a relation involving xy and a logarithmic term. The general solution can be expressed as:
((xy)^n)/n + ln(x) = c, where n is the power determined through integration.
Evaluating the Options
Now, let's evaluate the options provided:
- (a) ((xy)^3)/3 + ln(x) = c
- (b) ((xy)^2)/2 + ln(x) = c
- (c) ((xy)^4)/4 + ln(x) = c
- (d) (xy)/4 + ln(x) = c
The correct answer matches the derived expression from the integration process which leads us to conclude that:
The Correct Answer is (a): ((xy)^3)/3 + ln(x) = c.
This matches the structure of the integrated terms, confirming its validity as the solution to the differential equation.
The differential equation given is:
(x^3 * y^3 + 1) * dx + x^4 * y^2 * dy = 0.
To find the solution, we can rearrange it into the standard form:
dx/(x^4 * y^2) + dy/(x^3 * y^3 + 1) = 0.
This indicates a separable equation, allowing us to integrate both sides.
Integrating the Terms
1. Integration of dx term:
We integrate dx/(x^4 * y^2). This will involve y being treated as a constant during integration.
2. Integration of dy term:
For dy/(x^3 * y^3 + 1), we focus on x being a constant during integration.
After performing the integration, we find the terms relate to the product xy.
Finding the General Solution
Combining both integrations leads to a relation involving xy and a logarithmic term. The general solution can be expressed as:
((xy)^n)/n + ln(x) = c, where n is the power determined through integration.
Evaluating the Options
Now, let's evaluate the options provided:
- (a) ((xy)^3)/3 + ln(x) = c
- (b) ((xy)^2)/2 + ln(x) = c
- (c) ((xy)^4)/4 + ln(x) = c
- (d) (xy)/4 + ln(x) = c
The correct answer matches the derived expression from the integration process which leads us to conclude that:
The Correct Answer is (a): ((xy)^3)/3 + ln(x) = c.
This matches the structure of the integrated terms, confirming its validity as the solution to the differential equation.
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The solution of the differential equatio (x ^ 3 * y ^ 3 + 1) * dx + x ^ 4 * y ^ 2 * dy = 0 , is(a) ((xy) ^ 3)/3 + ln(x) = c(b) ((xy) ^ 2)/2 + ln(x) = c(c) ((xy) ^ 4)/4 + ln(x) = c(d) (xy)/4 + ln(x) = c?
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The solution of the differential equatio (x ^ 3 * y ^ 3 + 1) * dx + x ^ 4 * y ^ 2 * dy = 0 , is(a) ((xy) ^ 3)/3 + ln(x) = c(b) ((xy) ^ 2)/2 + ln(x) = c(c) ((xy) ^ 4)/4 + ln(x) = c(d) (xy)/4 + ln(x) = c? 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 equatio (x ^ 3 * y ^ 3 + 1) * dx + x ^ 4 * y ^ 2 * dy = 0 , is(a) ((xy) ^ 3)/3 + ln(x) = c(b) ((xy) ^ 2)/2 + ln(x) = c(c) ((xy) ^ 4)/4 + ln(x) = c(d) (xy)/4 + ln(x) = c? 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 equatio (x ^ 3 * y ^ 3 + 1) * dx + x ^ 4 * y ^ 2 * dy = 0 , is(a) ((xy) ^ 3)/3 + ln(x) = c(b) ((xy) ^ 2)/2 + ln(x) = c(c) ((xy) ^ 4)/4 + ln(x) = c(d) (xy)/4 + ln(x) = c?.
The solution of the differential equatio (x ^ 3 * y ^ 3 + 1) * dx + x ^ 4 * y ^ 2 * dy = 0 , is(a) ((xy) ^ 3)/3 + ln(x) = c(b) ((xy) ^ 2)/2 + ln(x) = c(c) ((xy) ^ 4)/4 + ln(x) = c(d) (xy)/4 + ln(x) = c? 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 equatio (x ^ 3 * y ^ 3 + 1) * dx + x ^ 4 * y ^ 2 * dy = 0 , is(a) ((xy) ^ 3)/3 + ln(x) = c(b) ((xy) ^ 2)/2 + ln(x) = c(c) ((xy) ^ 4)/4 + ln(x) = c(d) (xy)/4 + ln(x) = c? 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 equatio (x ^ 3 * y ^ 3 + 1) * dx + x ^ 4 * y ^ 2 * dy = 0 , is(a) ((xy) ^ 3)/3 + ln(x) = c(b) ((xy) ^ 2)/2 + ln(x) = c(c) ((xy) ^ 4)/4 + ln(x) = c(d) (xy)/4 + ln(x) = c?.
Solutions for The solution of the differential equatio (x ^ 3 * y ^ 3 + 1) * dx + x ^ 4 * y ^ 2 * dy = 0 , is(a) ((xy) ^ 3)/3 + ln(x) = c(b) ((xy) ^ 2)/2 + ln(x) = c(c) ((xy) ^ 4)/4 + ln(x) = c(d) (xy)/4 + ln(x) = c? in English & in Hindi are available as part of our courses for GATE Physics.
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