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

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

The given differential equation is
   $straight y space straight e to the power of straight x over straight y end exponent space dx space equals space open parentheses xe to the power of straight x over straight y end exponent plus straight y squared close parentheses space dy$
or $dx over dy space equals space fraction numerator xe to the power of begin display style straight x over straight y end style end exponent plus straight y squared over denominator ye to the power of begin display style straight x over straight y end style end exponent end fraction$

or $dx over dy space equals fraction numerator xe to the power of begin display style straight x over straight y end style end exponent plus straight y squared over denominator ye to the power of begin display style straight x over straight y end style end exponent end fraction$

or $dx over dy space equals space straight x over straight y plus straight y over straight e to the power of begin display style straight x over straight y end style end exponent$
Put y = v y so that $dx over dy equals straight v plus straight y dv over dx$
$therefore space space space space space space space space space straight v plus straight y dv over dy space equals space straight v plus ye to the power of negative straight v end exponent therefore space space space space space space space space straight y dv over dy space equals space straight y space straight e to the power of negative straight v end exponent space space space space space space space space space space space or space space space space space space dv over dy space equals straight e to the power of negative straight v end exponent$
Separting the variables and integration, we get,
   $integral straight e to the power of straight v dv space equals space integral dy space space space space space space space space space or space space space space space straight e to the power of straight v space equals space straight y plus straight c$
or $straight e to the power of straight x over straight y end exponent space equals space straight y plus straight c$