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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?
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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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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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