Find the orthogonal trajectories of the family of circles x² + (y - c)² = c², where c is a parameter.
To find the orthogonal trajectories of the given family of circles
\[
x^2 + (y - c)^2 = c^2
\]
where \( c \) is a parameter, we proceed as follows:
### Step 1: Differentiate the Given Family
Rewriting the given equation:
\[
x^2 + y^2 - 2cy + c^2 = __________ ________ _________ ______ ________ ________ ________.
____ ____ ____ _______ ________.
_____ ____ ________ ______ ___ _____ _______ _______ _________.
_______ ____ ___ ______ _______.
_________ _____ _______ __________ _________ __________ __________ __________.
_____ __________ _________ __________ _____ __________ __________ ________ _________ _____ _________ ________.
________ ______ _________ ________ _______ _________ _______ ________ ________ __________ __________ _________.
_______ _________ ____ ____ ___ ______ ___.
_______ _______ ___ __________ __________ _______ _________ ________ _________ __________ _________.
____ ________ __________ ________ ____.
____ ___ ______ ____ ___ _____ ____ _____ ________ ___.
_______ _________ ___ ____ ________ ___ ________ ______ ____ ____.
_________ _______ _______ ____ ____ _______ _______ ___.
_______ __________ __________ ___ ________.
____ _____ ________ _____ _________ _______ __________ ____ _____ __________.
_______ _____ ____ ___ ________ ______ _________ ____.
_________ ___ ___ _______ ________ __________ ___ _________ ____ ___ _________.
___ ____ ___ ___ ________ _________ ________ ____ ________ ______ _____ _____.
__________ ___ _______ ________ ______ ______ ________ _________ _____.
______ _________ _______ ________ ________ _____ _________ __________ __________ _________ ________.
_____ ____ ________ _______ ___ _________ _______ _________ _________.
_________ __________ ________ _____ _______.
________ ________ ____ _________ ___ ________.
__________ ___ ______ ______ _____ ___ _________ ________ __________.
___ _____ _____ ________ _______ _______ ___ _______ ____ ________ ________ ___.
_____ _____ _____ _________ _______ _____ ________ _________.
_____ ___ ______ __________ __________ _______ ______.
__________ ________ __________ ________ __________.
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xy" + y' + xy = 0, x > 0
into Bessel's equation and hence write its general solution in terms of Bessel's functions.
, and
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and
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.
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