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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'

The given differential equation is

straight y squared plus straight x squared dy over dx equals space straight x. space straight y space dy over dx

left parenthesis straight x space straight y space minus space straight x squared right parenthesis space dy over dx space equals straight y squared

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

Put

straight y space equals space straight v space straight x space so space that space dy over dx space equals straight v plus straight x dv over dx

therefore space space space space straight v plus straight x dv over dx equals space fraction numerator straight v squared minus straight x squared over denominator straight x squared straight v minus straight x squared end fraction therefore space space straight v plus straight x dv over dx space equals space fraction numerator straight v squared over denominator straight v minus 1 end fraction therefore space space space space space space space space straight x dv over dx space equals fraction numerator straight v squared over denominator straight v minus 1 end fraction minus straight v therefore space space space space space space straight x dv over dx space equals space fraction numerator straight v over denominator straight v minus 1 end fraction therefore space space space space space fraction numerator straight v minus 1 over denominator straight v end fraction dv space equals space 1 over straight x dx therefore space space space space integral open parentheses 1 minus 1 over straight v close parentheses space dv space equals space integral 1 over straight x dx therefore space space space space space straight v minus log space straight v space equals space logx plus log space straight c therefore space space space space straight y over straight x minus log space straight y over straight x space equals space log space straight x space plus space log space straight c therefore space space space space straight y over straight x minus log space straight y space equals space log space straight c 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 space space space space space space space space space space space space space space space space rightwards double arrow space space straight y over straight x space equals space log space straight c space straight y therefore space space space space space space space space space straight c space straight y space equals space straight e to the power of straight y over straight x end exponent space is space the space required space solution. space

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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

76 Views

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

75 Views

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

72 Views

Short Answer Type

216.

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

82 Views

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

72 Views

Short Answer Type

219.

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