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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

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

The given differential equation is

open parentheses straight x space sin space straight y over straight x close parentheses dy space equals space open parentheses ysin straight y over straight x minus straight x close parentheses dx space space space space space space or space space space space dy over dx equals fraction numerator ysin begin display style straight y over straight x end style minus straight x over denominator straight x space sin begin display style straight y over straight x end style end fraction space space space space space... left parenthesis 1 right parenthesis

Put y = v x so that

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

therefore space space left parenthesis 1 right parenthesis space becomes comma space space straight v plus straight x dv over dx equals fraction numerator straight v space straight x space sin space straight v space minus space straight x over denominator straight x space sin space straight v end fraction

straight v plus straight x dv over dx equals fraction numerator straight v space sinv minus 1 over denominator sin space straight v end fraction space space space or space space space straight x dv over dx equals fraction numerator straight v space sinv minus 1 over denominator sin space straight v end fraction minus straight v space space space or space space straight x dv over dx equals negative fraction numerator 1 over denominator sin space straight v end fraction

Separating the variables,

sin space straight v space dv space equals space minus 1 over straight x dx

Integrating,

negative cos space straight v space equals space minus log space open vertical bar straight x close vertical bar space minus space straight c space space space or space cos space open parentheses straight y over straight x close parentheses space equals space log space open vertical bar straight x close vertical bar space plus straight c

which is required solution.

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