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Vector Algebra

Question
CBSEENMA12033019

Solve the following differential equation:
dy over dx space equals space log space left parenthesis straight x plus 1 right parenthesis

Solution
The given differential equation is
                      dy over dx space equals space log space left parenthesis straight x plus 1 right parenthesis
Separating the variables, we get,
                        dy space equals space log space left parenthesis straight x plus 1 right parenthesis. space dx
Integrating,       integral dy space equals space integral log space left parenthesis straight x plus 1 right parenthesis. space 1 space dx
therefore space space space space space space space space space space space straight y space equals space log space left parenthesis straight x plus 1 right parenthesis space. straight x space space minus space integral fraction numerator 1 over denominator straight x plus 1 end fraction straight x space dx
or            straight y equals straight x space log left parenthesis straight x plus 1 right parenthesis minus space integral fraction numerator left parenthesis straight x plus 1 right parenthesis space minus space 1 over denominator straight x plus 1 end fraction dx
or            straight y space equals straight x space log space left parenthesis straight x plus 1 right parenthesis space minus space integral open parentheses 1 minus fraction numerator 1 over denominator straight x plus 1 end fraction close parentheses dx
or           straight y equals straight x space log space left parenthesis straight x plus 1 right parenthesis space minus space open square brackets straight x minus log left parenthesis straight x plus 1 right parenthesis close square brackets plus straight c
or          straight y equals space straight x space log space left parenthesis straight x plus 1 right parenthesis space minus space straight x space plus space log space left parenthesis straight x plus 1 right parenthesis space plus straight c
or           straight y equals left parenthesis straight x plus 1 right parenthesis space log space left parenthesis straight x plus 1 right parenthesis space minus straight x space plus straight c
which is required solution.