Show that the given differential equation is homogeneous and solve it. (x 2 – y 2 ) dx + 2xy dy = 0 given that y = 1 when x = 1. - Wired Faculty

Question

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Show that the given differential equation is homogeneous and solve it.
(x² – y²) dx + 2xy dy = 0
given that y = 1 when x = 1.

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Solution

The given differential equation is

open parentheses straight x squared minus straight y squared close parentheses space dx space plus space 2 xy space dy space equals space 0 space space space space or space space space 2 xy space dy space equals space left parenthesis straight y squared minus straight x squared right parenthesis space dx

dy over dx space equals space fraction numerator straight y squared minus straight x squared over denominator 2 xy end fraction

Put y = vx so that

dy over dx equals space straight v plus straight x dv over dx

therefore space space space space straight v plus straight x dv over dx equals fraction numerator straight v squared straight x squared minus straight x squared over denominator 2 vx squared end fraction space space space or space space space straight v plus straight x dv over dx space equals space fraction numerator straight v squared minus 1 over denominator 2 space straight v end fraction

therefore space space space space space space space space space space space straight x dv over dx equals fraction numerator straight v squared minus 1 over denominator 2 space straight v end fraction minus straight v space space space or space space space straight x dv over dx space equals space fraction numerator straight v squared minus 1 minus 2 straight v squared over denominator 2 straight v end fraction therefore space space space space space space space space straight x dv over dx space equals space fraction numerator negative 1 minus straight v squared over denominator 2 space straight v end fraction space space space space space rightwards double arrow space space space space space space space fraction numerator 2 space straight v over denominator 1 plus straight v squared end fraction dv space equals space minus 1 over straight x dx therefore space space space space space space integral fraction numerator 2 space straight v over denominator 1 plus straight v squared end fraction dv space equals space minus integral 1 over straight x dx therefore space space space log space open vertical bar 1 plus straight v squared close vertical bar space equals space minus log space open vertical bar straight x close vertical bar plus straight c apostrophe therefore space space space log space open vertical bar 1 plus straight v squared close vertical bar plus log space open vertical bar straight x close vertical bar space equals space straight c apostrophe therefore space log space open vertical bar left parenthesis 1 plus straight v squared right parenthesis space left parenthesis straight x right parenthesis close vertical bar space equals space straight c apostrophe therefore space space space space space space straight x left parenthesis 1 plus straight v squared right parenthesis space equals space straight c apostrophe space space space space space space space rightwards double arrow space space space space space straight x space open parentheses 1 plus straight y squared over straight x squared close parentheses space equals space straight c therefore space space space space space space straight x squared plus straight y squared space equals space straight c space straight x

Now,

straight y space equals space 1 comma space space when space straight x space equals space 1

therefore space space space space space space space space space space 1 plus 1 space equals space straight c space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight c space equals space 2 therefore space space space solution space is space straight x squared plus straight y squared space equals space 2 straight x.

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Vector Algebra

Show that the given differential equation is homogeneous and solve it.
(x² – y²) dx + 2xy dy = 0
given that y = 1 when x = 1.

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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NCERT Sample Papers

Entrance Exams Preparation

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For Daily Free Study Material Join wiredfaculty

Vector Algebra

Show that the given differential equation is homogeneous and solve it.(x2 – y2) dx + 2xy dy = 0given that y = 1 when x = 1.

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

Show that the given differential equation is homogeneous and solve it.
(x² – y²) dx + 2xy dy = 0
given that y = 1 when x = 1.

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