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Vector Algebra

Question
CBSEENMA12032899

For problem given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation:
straight x squared space equals space 2 straight y squared space log space straight y    :     left parenthesis straight x squared plus straight y squared right parenthesis space dy over dx minus straight x space straight y space equals space 0
   

Solution
The given differential equation is
                            straight x squared space equals space 2 straight y squared log space straight y                 ...(1)
therefore space space space space space         2 straight x space equals space 2 open curly brackets straight y squared cross times 1 over straight y dy over dx plus space log space straight y space open parentheses 2 space straight y space dy over dx close parentheses close curly brackets
or                  2 straight x space equals 2 left parenthesis straight y plus 2 space straight y space log space straight y right parenthesis dy over dx
rightwards double arrow               straight x equals left parenthesis straight y plus 2 space straight y space log space straight y right parenthesis dy over dx
Multiplying both sides by y, we get
                                straight x space straight y space equals space left parenthesis straight y squared plus 2 straight y squared space log space straight y right parenthesis dy over dx
or                            xy space equals space left parenthesis straight y squared plus straight x squared right parenthesis space dy over dx                   open square brackets because space of space left parenthesis 1 right parenthesis close square brackets
or          left parenthesis straight x squared plus straight y squared right parenthesis space dy over dx minus straight x space straight y space equals space 0
Hence the result.