Differential equation with separable, probably wrong answer in book I have a differential equation: (dy)/(dx) = y log(y)cot(x)

Aryanna Blake 2022-10-26 Answered
Differential equation with separable, probably wrong answer in book
I have a differential equation:
d y d x = y log ( y ) cot ( x )
I'm trying solve that equation by separating variables and dividing by y log ( y )
d y = y log ( y ) cot ( x ) d x
d y y log ( y ) = cot ( x ) d x
cot ( x ) d y y log ( y ) = 0
Where of course y>0 regarding to division
Beacuse:
d y y log ( y ) = ln | ln ( y ) | + C
and:
cot ( x ) d x = ln | sin ( x ) | + C
So:
ln | sin ( x ) | ln | ln ( y ) | = C
ln | sin ( x ) ln ( y ) | = C
sin ( x ) ln ( y ) = ± e C
d = ± e C
sin ( x ) = d ln ( y )
sin ( x ) d = ln ( y )
e sin ( x ) d = y
This is my final answer. I have problem because in book from equation comes the answer to exercise is:
y = e c sin ( x )
Which one is correct?
I will be grateful for explaining Best regards
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Answers (1)

indivisast7
Answered 2022-10-27 Author has 13 answers
Note that
d u u = ln | u | + c 1 = ln | u | + ln | c | = ln | c u | .
Starting from your result:
cot x d x = d y y ln y
d ( sin x ) sin x = d ( ln y ) y ln y
we integrate both sides:
d ( sin x ) sin x d x = d ( ln y ) ln y d x
ln | a sin x | = ln | b ln y |
a sin x = b ln y , ( 1 )
ln y = c sin x
y = e c sin x
where a,b,c are arbitrary constants. Note that (1) includes all possible solutions including the case a=0.
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