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The solution of the equation cosx cosy (dy/dx)=-sinx siny is
- a)siny+cosx=c
- b)siny-cosx=c
- c)siny-c cosx=c
- d)siny=c cosx
Correct answer is option 'D'. Can you explain this answer?
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The solution of the equation cosx cosy (dy/dx)=-sinx siny isa)siny+cos...
Solution:
Given equation is cosx cosy (dy/dx)=-sinx siny
We need to solve for y, so we need to separate the variables x and y.
Separating the variables, we get:
(dy/dx) = (-sinx siny) / (cosx cosy)
(dy/dx) = (-sinx/cosx) * (siny/cosy)
(dy/dx) = (-tanx) * (tan y)
Integrating both sides, we get:
∫ (1/y) dy = ∫ (-tanx) dx
ln |y| = ln |cosx| + c1
|y| = e^(ln |cosx| + c1) = e^c1 * cosx
Since e^c1 is just another constant, we can write:
|y| = C * cosx
where C is a constant of integration.
Now, we need to determine the sign of y. To do this, we look at the original equation:
cosx cosy (dy/dx)=-sinx siny
We see that if sinx and siny have the same sign (both positive or both negative), then dy/dx must be negative (since cosx and cosy are always positive). Therefore, y must be decreasing as x increases, which means that y is negative.
Thus, we can write:
-y = C * cosx
Multiplying both sides by -1, we get:
y = -C * cosx
or
y = C' * sinx
where C' = -C.
Therefore, the solution of the equation is:
y = C' * sinx
or
siny = C cosx
The correct option is (d), siny = C cosx.
Given equation is cosx cosy (dy/dx)=-sinx siny
We need to solve for y, so we need to separate the variables x and y.
Separating the variables, we get:
(dy/dx) = (-sinx siny) / (cosx cosy)
(dy/dx) = (-sinx/cosx) * (siny/cosy)
(dy/dx) = (-tanx) * (tan y)
Integrating both sides, we get:
∫ (1/y) dy = ∫ (-tanx) dx
ln |y| = ln |cosx| + c1
|y| = e^(ln |cosx| + c1) = e^c1 * cosx
Since e^c1 is just another constant, we can write:
|y| = C * cosx
where C is a constant of integration.
Now, we need to determine the sign of y. To do this, we look at the original equation:
cosx cosy (dy/dx)=-sinx siny
We see that if sinx and siny have the same sign (both positive or both negative), then dy/dx must be negative (since cosx and cosy are always positive). Therefore, y must be decreasing as x increases, which means that y is negative.
Thus, we can write:
-y = C * cosx
Multiplying both sides by -1, we get:
y = -C * cosx
or
y = C' * sinx
where C' = -C.
Therefore, the solution of the equation is:
y = C' * sinx
or
siny = C cosx
The correct option is (d), siny = C cosx.
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The solution of the equation cosx cosy (dy/dx)=-sinx siny isa)siny+cosx=cb)siny-cosx=cc)siny-c cosx=cd)siny=c cosxCorrect answer is option 'D'. Can you explain this answer?
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