What is a solution to the differential equation dy/dx=2csc^2x/cotx?

rustenig 2022-09-14 Answered
What is a solution to the differential equation d y d x = 2 csc 2 x cot x ?
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Answers (2)

Ashlee Ramos
Answered 2022-09-15 Author has 20 answers
The equation simplifies:
d y d x = 2 csc ( x ) sec ( x )
Separate variables:
d y = 2 csc ( x ) sec ( x ) d x
Integrate:
d y = 2 csc ( x ) sec ( x ) d x
y = 2 ln | sin ( x ) | - 2 ln | cos ( x ) | + C
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equipokypip1
Answered 2022-09-16 Author has 5 answers
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First we should separate the variables, which means that we can treat d y d x like division. We can move the dx to the right hand side of the equation to be with all the other terms including x.
d y = 2 csc 2 x cot x d x
Now integrate both sides:
d y = 2 csc 2 x cot x d x
On the right hand side, let u=cotx. This implies that d u = - csc 2 x d x .
y = - 2 - csc 2 x cot x d x
y = - 2 d u u
y = - 2 ln | u | + C
y = - 2 ln | cot x | + C
One possible simplification we could make if we wanted would be to bring the −1 outside the logarithm into the logarithm as a −1 power:
y = 2 ln | tan x | + C
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