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Differential Equations

Short Answer Type

211. Find a one parameter family of solutions of each of the following differential equation:
y² + x² y' = x y y'

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Long Answer Type

212. Find a one parameter family of solutions of each of the following differential equation:
(y² – 2xy) dx = (x² – 2xy) dy

The given differential equation is
(y² – 2xy) dx = (x² – 2xy) dy
or

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

Put y = v x so that

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

therefore space space space space space space space space straight v plus straight x dv over dx space equals space fraction numerator straight v squared straight x squared minus 2 vx squared over denominator straight x squared minus 2 vx squared end fraction therefore space space space space space space space straight v plus straight x dv over dx space equals fraction numerator straight v squared minus 2 straight v over denominator 1 minus 2 straight v end fraction therefore space space space space space space space space space space straight x dv over dx space equals space fraction numerator straight v squared minus 2 straight v over denominator 1 minus 2 straight v end fraction minus straight v space equals space fraction numerator straight v squared minus 2 straight v minus straight v plus 2 straight v squared over denominator 1 minus 2 straight v end fraction therefore space space space space space space space straight x dv over dx space equals space fraction numerator 3 straight v squared minus 3 straight v over denominator 1 minus 2 straight v end fraction

Separating the variables and integrating, we get,

integral fraction numerator 1 minus 2 straight v over denominator 3 straight v squared minus 3 straight v end fraction equals space space integral 1 over straight x dx space space or space space 1 third integral fraction numerator 1 minus 2 straight v over denominator straight v squared minus straight v end fraction dv space equals integral 1 over straight x dx

therefore space space space space minus 1 third space log space left parenthesis straight v squared minus straight v right parenthesis space equals space log space straight x plus space log space straight c therefore space space space space log space left parenthesis straight v squared minus straight v right parenthesis to the power of fraction numerator negative 1 over denominator 3 end fraction end exponent space equals space log space cx space therefore space space space space space space space space space space space left parenthesis straight v squared minus straight v right parenthesis to the power of negative 1 third end exponent space equals space straight c space straight x therefore space space space space space space space open parentheses straight y squared over straight x squared minus straight y over straight x close parentheses to the power of negative 1 third end exponent space equals space space straight c space straight x therefore space space space space space space open parentheses straight y squared minus straight x space straight y close parentheses to the power of negative 1 third end exponent space equals space cx space cross times space straight x to the power of fraction numerator negative 2 over denominator 3 end fraction end exponent therefore space space space space space space space space left parenthesis straight y squared minus straight x space straight y right parenthesis to the power of fraction numerator negative 1 over denominator 3 end fraction end exponent space equals space straight c space straight x to the power of 1 third end exponent therefore space space space space space space space space space left parenthesis straight y squared minus straight x space straight y right parenthesis to the power of negative 1 end exponent space equals space Ax therefore space space space space space space fraction numerator 1 over denominator straight y squared minus xy end fraction space equals space Ax or space space space xy squared minus straight x squared straight y space equals space straight c comma space which space is space required space solution. space

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

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

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215. Solve the following differential equation:

open parentheses 1 plus straight e to the power of straight x over straight y end exponent close parentheses space dx space plus space straight e to the power of straight x over straight y end exponent space open parentheses 1 minus straight x over straight y close parentheses dy space equals space 0

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Short Answer Type

216.

Solve $open parentheses straight x space sin space straight y over straight x close parentheses dy space equals open parentheses straight y space sin space straight y over straight x minus straight x close parentheses dx$

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Long Answer Type

217.

Show that the differential equation $straight x space cos space open parentheses straight y over straight x close parentheses dy over dx space equals straight y space cos space open parentheses straight y over straight x close parentheses plus straight x$ is homogeneous and solve it.

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218.

Show that the differential equation:
$left parenthesis straight x space dy space minus space straight y space dx right parenthesis space straight y space sin space open parentheses straight y over straight x close parentheses space equals space left parenthesis straight y space dx space plus space straight x space dy right parenthesis space straight x space cos space open parentheses straight y over straight x close parentheses$

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Short Answer Type

219.

Solve:
$straight y space straight e to the power of straight x over straight y end exponent dx space equals open parentheses xe to the power of straight x over straight y end exponent plus straight y squared close parentheses space space dy comma space space space straight y not equal to 0$