Question

Prove that x² – y² = c (x² + y² )² is the general solution of differential equation (x³ – 3 x y² ) dx = (y³ –3 x² y) dy . where c is a parameter.

Solution

Here, $straight x squared minus straight y squared space equals space straight c left parenthesis straight x squared plus straight y squared right parenthesis squared$ ...(1)
Differentiating w.r.t.x, we get,
$2 straight x minus 2 straight y dy over dx space equals space 2 straight c left parenthesis straight x squared plus straight y squared right parenthesis. space open parentheses 2 straight x plus 2 straight y dy over dx close parentheses$
$straight x minus straight y space dy over dx space equals space 2 straight c left parenthesis straight x squared plus straight y squared right parenthesis. space open parentheses straight x plus straight y dy over dx close parentheses$ ...(2)
Dividing (2) by (1), we get.
$fraction numerator straight x minus straight y begin display style dy over dx end style over denominator straight x squared minus straight y squared end fraction space equals space fraction numerator 2 open parentheses straight x plus straight y begin display style dy over dx end style close parentheses over denominator straight x squared plus straight y squared end fraction$
$therefore space space space straight x left parenthesis straight x squared plus straight y squared right parenthesis minus straight y left parenthesis straight x squared plus straight y squared right parenthesis space dy over dx space equals space 2 straight x left parenthesis straight x squared minus straight y squared right parenthesis plus 2 straight y left parenthesis straight x squared minus straight y squared right parenthesis dy over dx therefore space space space space space open square brackets 2 space straight y space left parenthesis straight x squared minus straight y squared right parenthesis space plus space straight y left parenthesis straight x squared plus straight y squared right parenthesis close square brackets dy over dx space equals space straight x left parenthesis straight x squared plus straight y squared right parenthesis minus 2 straight x left parenthesis straight x squared minus straight y squared right parenthesis therefore space space space left parenthesis 2 straight x squared straight y minus 2 straight y cubed plus straight x squared straight y plus straight y cubed right parenthesis dy over dx space equals space straight x cubed plus xy squared minus 2 straight x cubed plus 2 xy squared therefore space space space space left parenthesis 3 straight x squared straight y minus straight y cubed right parenthesis dy over dx space equals space 3 xy squared minus straight x cubed therefore space space space space space space space space space space space space space space space space space space space space space space space space space space space dy over dx space equals space fraction numerator straight x cubed minus 3 xy squared over denominator straight y cubed minus 3 straight x squared straight y end fraction space which space is space the space required space equation.$
Hence the result.

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Vector Algebra

Prove that x² – y² = c (x² + y² )² is the general solution of differential equation (x³ – 3 x y² ) dx = (y³ –3 x² y) dy . where c is a parameter.

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

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Vector Algebra

Prove that x2 – y2 = c (x2 + y2 )2 is the general solution of differential equation (x3 – 3 x y2 ) dx = (y3 –3 x2 y) dy . where c is a parameter.

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

Prove that x² – y² = c (x² + y² )² is the general solution of differential equation (x³ – 3 x y² ) dx = (y³ –3 x² y) dy . where c is a parameter.

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