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The solution of the differential equation cos x sin y dx + sin x cos y dy =0 is
- a)sin x/sin y = C
- b)cos x + cos y =C
- c)sin x + sin y = C
- d)sin x sin y = C
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The solution of the differential equation cos x sin y dx + sin x cos y...
Mansi
first of all separate each term of x and y on both side .
you will form of Cotx and Cot y on both side .
then Integrate them .
now apply property of log multiplication.
you will get answer
thanks
first of all separate each term of x and y on both side .
you will form of Cotx and Cot y on both side .
then Integrate them .
now apply property of log multiplication.
you will get answer
thanks
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The solution of the differential equation cos x sin y dx + sin x cos y...
Solution:
Given differential equation: cos(x)sin(y)dx - sin(x)cos(y)dy = 0
Separating the variables:
cos(x)sin(y)dx = sin(x)cos(y)dy
Dividing both sides by sin(x)cos(y):
(dx/dy) = sin(x)/sin(y)
Integrating both sides with respect to x:
∫dx = ∫sin(x)/sin(y) dy
The integral of dx is simply x, and the integral of sin(x)/sin(y) dy can be solved using trigonometric identities.
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite sin(x)/sin(y) as:
sin(x)/sin(y) = √(1 - cos^2(x))/sin(y) = √(1 - (sin^2(y)))/sin(y) = √(1 - sin^2(y))/sin(y) = √(cos^2(y))/sin(y) = cos(y)/sin(y)
Integrating both sides:
x = ∫(cos(y)/sin(y)) dy
This integral can be solved by substituting u = sin(y), du = cos(y)dy:
x = ∫(1/u) du
x = ln|u| + C
x = ln|sin(y)| + C
Therefore, the solution to the given differential equation is:
ln|sin(y)| + C = x
Rearranging the equation:
ln|sin(y)| = x - C
Taking the exponential of both sides:
|sin(y)| = e^(x - C)
Since the absolute value of sin(y) can take positive or negative values, we can remove the absolute value sign:
sin(y) = ±e^(x - C)
This can be further simplified as:
sin(y) = Ce^x
where C = ±e^(-C)
Therefore, the correct solution to the given differential equation is:
sin(x)sin(y) = C
Given differential equation: cos(x)sin(y)dx - sin(x)cos(y)dy = 0
Separating the variables:
cos(x)sin(y)dx = sin(x)cos(y)dy
Dividing both sides by sin(x)cos(y):
(dx/dy) = sin(x)/sin(y)
Integrating both sides with respect to x:
∫dx = ∫sin(x)/sin(y) dy
The integral of dx is simply x, and the integral of sin(x)/sin(y) dy can be solved using trigonometric identities.
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite sin(x)/sin(y) as:
sin(x)/sin(y) = √(1 - cos^2(x))/sin(y) = √(1 - (sin^2(y)))/sin(y) = √(1 - sin^2(y))/sin(y) = √(cos^2(y))/sin(y) = cos(y)/sin(y)
Integrating both sides:
x = ∫(cos(y)/sin(y)) dy
This integral can be solved by substituting u = sin(y), du = cos(y)dy:
x = ∫(1/u) du
x = ln|u| + C
x = ln|sin(y)| + C
Therefore, the solution to the given differential equation is:
ln|sin(y)| + C = x
Rearranging the equation:
ln|sin(y)| = x - C
Taking the exponential of both sides:
|sin(y)| = e^(x - C)
Since the absolute value of sin(y) can take positive or negative values, we can remove the absolute value sign:
sin(y) = ±e^(x - C)
This can be further simplified as:
sin(y) = Ce^x
where C = ±e^(-C)
Therefore, the correct solution to the given differential equation is:
sin(x)sin(y) = C
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The solution of the differential equation cos x sin y dx + sin x cos y dy =0 isa)sin x/sin y = Cb)cos x + cos y =Cc)sin x + sin y = Cd)sin x sin y = CCorrect answer is option 'D'. Can you explain this answer?
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The solution of the differential equation cos x sin y dx + sin x cos y dy =0 isa)sin x/sin y = Cb)cos x + cos y =Cc)sin x + sin y = Cd)sin x sin y = CCorrect answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The solution of the differential equation cos x sin y dx + sin x cos y dy =0 isa)sin x/sin y = Cb)cos x + cos y =Cc)sin x + sin y = Cd)sin x sin y = CCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The solution of the differential equation cos x sin y dx + sin x cos y dy =0 isa)sin x/sin y = Cb)cos x + cos y =Cc)sin x + sin y = Cd)sin x sin y = CCorrect answer is option 'D'. Can you explain this answer?.
The solution of the differential equation cos x sin y dx + sin x cos y dy =0 isa)sin x/sin y = Cb)cos x + cos y =Cc)sin x + sin y = Cd)sin x sin y = CCorrect answer is option 'D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The solution of the differential equation cos x sin y dx + sin x cos y dy =0 isa)sin x/sin y = Cb)cos x + cos y =Cc)sin x + sin y = Cd)sin x sin y = CCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The solution of the differential equation cos x sin y dx + sin x cos y dy =0 isa)sin x/sin y = Cb)cos x + cos y =Cc)sin x + sin y = Cd)sin x sin y = CCorrect answer is option 'D'. Can you explain this answer?.
Solutions for The solution of the differential equation cos x sin y dx + sin x cos y dy =0 isa)sin x/sin y = Cb)cos x + cos y =Cc)sin x + sin y = Cd)sin x sin y = CCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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