Find a one parameter family of solutions of each of the following differential equation:
(x y² + 2x) dx + (x² y + 2y) dy = 0

Solution

The given differential equation is
(x y² + 2x) dx + (x² y + 2y) dy = 0
or $straight y left parenthesis straight x squared plus 2 right parenthesis space dx space equals space minus straight x left parenthesis straight y squared plus 2 right parenthesis space dx$
$therefore space space space space space fraction numerator straight y over denominator straight y squared plus 2 end fraction dy space equals negative fraction numerator straight x over denominator straight x squared plus 2 end fraction dx$
Integrating, $integral fraction numerator 2 straight y over denominator straight y squared plus 2 end fraction dy space equals negative integral fraction numerator 2 straight x over denominator straight x squared plus 2 end fraction dx$
$therefore space space space space space space log space left parenthesis straight y squared plus 2 right parenthesis space equals space minus log left parenthesis straight x squared plus 2 right parenthesis space plus space log space straight c space space space or space space space log space open square brackets left parenthesis straight y squared plus 2 right parenthesis space left parenthesis straight x squared plus 2 right parenthesis close square brackets space plus space log space straight c therefore space space space space left parenthesis straight y squared plus 2 right parenthesis thin space left parenthesis straight x squared plus 2 right parenthesis space equals space straight c space is space the space required space solution.$

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Vector Algebra

Find a one parameter family of solutions of each of the following differential equation:
(x y² + 2x) dx + (x² y + 2y) dy = 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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Vector Algebra

Find a one parameter family of solutions of each of the following differential equation: (x y2 + 2x) dx + (x2 y + 2y) dy = 0

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

Find a one parameter family of solutions of each of the following differential equation:
(x y² + 2x) dx + (x² y + 2y) dy = 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