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Differential Equations

Short Answer Type

221.

Solve:
$straight y apostrophe space equals space straight y over straight x plus sin straight y over straight x$

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222.
Solve:
$dy over dx space equals straight y over straight x plus tan straight y over straight x open parentheses 0 less than straight y over straight x less than straight pi over 2 close parentheses$

The given differential equation is
$dy over dx space equals space straight y over straight x plus tan straight y over straight x space space space space space space space space space space space space open parentheses 0 less than straight y over straight x less than straight pi over 2 close parentheses$ ...(1)
Put y = v x so that $dy over dx equals straight v plus straight x dv over dx$
$therefore space space space space from space left parenthesis 1 right parenthesis comma space space straight v plus straight x dv over dx space equals space straight v plus tan space straight v space space or space space space straight x dv over dx space equals space tan space straight v space space or space space space fraction numerator 1 over denominator tan space straight v end fraction dv space equals space dx over straight x$

or $integral cot space straight v space dv space equals space integral dx over straight x$

or $log space open vertical bar sin space straight v close vertical bar space equals space log space open vertical bar straight x close vertical bar space plus space log space open vertical bar straight c close vertical bar space space or space space space log space open vertical bar sin space straight v close vertical bar space equals space log space open vertical bar cx close vertical bar$

or $sin space straight v space equals space cx space space space space or space space space space space sin straight y over straight x space equals space space straight c space straight x space is space the space required space solution. space$

74 Views

Long Answer Type

223.

For the given differential equations, find the particular solution satisfying the given condition:
$left parenthesis straight x plus straight y right parenthesis space dy space plus space left parenthesis straight x minus straight y right parenthesis dx space equals space 0 space semicolon space space straight y space equals space 1 space space space when space straight x space equals space 1$

75 Views

224.

For the given differential equation, find the particular solution satisfying the given condition:
$straight x squared dy plus left parenthesis xy plus straight y squared right parenthesis dx space equals space 0 semicolon space space space straight y space equals space 1 space when space straight x space equals space 1$

80 Views

225.

For the given differential equation, find the particular solution satisfying the given condition:
$open square brackets straight x space sin squared open parentheses straight y over straight x close parentheses minus straight y close square brackets space dx plus space straight x space dy space equals space 0 colon space space space straight y space equals straight pi over 4 space space when space straight x space equals space 1$

73 Views

226.

For the given differential equation, find the particular solution satisfying the given condition:
$dy over dx minus straight y over straight x plus cosec space open parentheses straight y over straight x close parentheses space equals space 0 space semicolon space space space straight y space equals space 0 space space space when space straight x space equals space 1$

75 Views

227.

For the given differential equation, find the particular solution satisfying the given condition:
$2 xy plus straight y squared minus 2 straight x squared dy over dx equals 0 semicolon space space space straight y space equals 2 space space when space straight x space equals space 1.$

75 Views

228.

For the given differential equation, find the particular solution satisfying the given condition:
$2 straight x squared straight y apostrophe space minus space 2 xy plus straight y squared space equals space 0 comma space space space space straight y left parenthesis straight e right parenthesis space equals space straight e$

74 Views

229.

For the given differential equation, find the particular solution satisfying the given condition:
$2 xy plus straight y squared minus 2 straight x squared straight y apostrophe space equals space 0 space space space space space space space space space space space space space straight y left parenthesis 1 right parenthesis space equals space 2$