Question

Solve:
$left parenthesis 1 plus straight y squared right parenthesis thin space left parenthesis 1 plus log space straight x right parenthesis space dx space plus space straight x space dy space equals space 0$

Solution

The given differential equation is
$left parenthesis 1 plus straight y squared right parenthesis thin space left parenthesis 1 plus log space straight x right parenthesis space dx space plus space straight x space dy space equals space 0 space space or space space straight x space dy space equals space minus left parenthesis 1 plus straight y squared right parenthesis thin space left parenthesis 1 plus space log space straight x right parenthesis space dx$
Separating the variables, we get,
$fraction numerator dy over denominator 1 plus straight y squared end fraction space equals negative fraction numerator 1 plus log space straight x over denominator straight x end fraction dx$
Integrating, $integral fraction numerator 1 over denominator 1 plus straight y squared end fraction dy space equals space minus integral left parenthesis 1 plus log space straight x right parenthesis. space 1 over straight x dx$
$therefore space space space tan to the power of negative 1 end exponent straight y space plus straight c space equals space minus fraction numerator left parenthesis 1 plus space log space straight x right parenthesis squared over denominator 2 end fraction space space space space space open square brackets because space space integral open square brackets straight f left parenthesis straight x right parenthesis close square brackets to the power of straight n space straight f apostrophe left parenthesis straight x right parenthesis space dx space equals space fraction numerator straight f to the power of straight n plus 1 end exponent left parenthesis straight x right parenthesis over denominator straight n plus 1 end fraction plus straight c close square brackets therefore space space space tan to the power of negative 1 end exponent straight y space plus space 1 half left parenthesis 1 plus space log space straight x right parenthesis squared space plus space straight c space equals space 0$
which is the required solution.

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Vector Algebra

Solve:
$left parenthesis 1 plus straight y squared right parenthesis thin space left parenthesis 1 plus log space straight x right parenthesis space dx space plus space straight x space dy space equals space 0$

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

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Loaction

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Vector Algebra

Solve:

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:y' + y = ex

Determine order and degree (if defined) of differential equations:y'' + (y')2 + 2 y = 0

Determine order and degree (if defined) of differential equations:y″ + 2y′ + sin y = 0

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Solve:
$left parenthesis 1 plus straight y squared right parenthesis thin space left parenthesis 1 plus log space straight x right parenthesis space dx space plus space straight x space dy space equals space 0$

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0