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Differential Equations

Short Answer Type

141.

Solve:
$open parentheses straight e to the power of straight x plus 1 close parentheses space straight y space dy space plus space left parenthesis straight y plus 1 right parenthesis space straight e to the power of straight x space dx space equals space 0$

87 Views

142.

Solve:
$straight y left parenthesis 1 minus straight x squared right parenthesis space dy space plus space straight x space left parenthesis 1 plus straight y squared right parenthesis space dx space equals space 0.$

83 Views

143.

Solve:
$straight y space logydx space minus space straight x space dy space equals space 0$

91 Views

Long Answer Type

144.

Show that the general solution of the differential equation $dy over dx plus fraction numerator straight y squared plus straight y plus 1 over denominator straight x squared plus straight x plus 1 end fraction space equals 0$ is given by (x + y + 1) = A (1 – x – y – 2 x y), where A is parameter.

77 Views

Short Answer Type

145.
Solve:
$dy over dx space equals space fraction numerator straight x left parenthesis 2 space log space straight x space plus space 1 right parenthesis over denominator sin space straight y space plus space straight y space cosy end fraction$

The given differential equation is
$dy over dx space equals space fraction numerator straight x left parenthesis 2 space log space straight x space plus space 1 right parenthesis over denominator sin space straight y space plus space straight y space cosy end fraction$
Separating the variables, we get,
$left parenthesis sin space straight y space plus space straight y space cosy right parenthesis space dy space equals space straight x left parenthesis 2 space logx space plus 1 right parenthesis space dx$
$therefore space space space space integral left parenthesis sin space straight y space plus straight y space cos right parenthesis space dy space equals space integral straight x space left parenthesis 2 space log space straight x space plus 1 right parenthesis space dx therefore space space space integral sin space ydy space plus space integral straight y space cosy space dy space equals space 2 space integral logx. space straight x space dx space plus space integral straight x space dx therefore space space space space space space space minus cosy space plus space straight y space siny space minus space integral 1. space sin space straight y space dy space equals space 2 open square brackets left parenthesis log space straight x right parenthesis. space straight x squared over 2 minus integral 1 over straight x. straight x squared over 2 dx plus straight x squared over 2 close square brackets therefore space space space minus cosy space plus space ysiny space space plus space cosy space space equals space straight x squared logx minus straight x squared over 2 plus straight x squared over 2 plus straight c therefore space space space straight y space sin space straight y space equals space straight x squared space logx space plus straight c space which space is space required space equation.$

80 Views

146.

Solve
$dy over dx space equals space fraction numerator xe to the power of straight x logx plus straight e to the power of straight x over denominator straight x space cosy end fraction$

83 Views

Long Answer Type

147.

Solve
$dy over dx space equals space sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x$

82 Views

148.

Solve:
$dy over dx space equals space cos cubed straight x space sin to the power of 4 straight x plus straight x square root of 2 straight x plus 1 end root$

84 Views

149.

Solve:
$dy over dx equals negative straight x space sin squared straight x space space equals space fraction numerator 1 over denominator straight x space log space straight x end fraction$