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

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

The given differential equation is
   $dy over dx space equals space sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x$
Separting the variables, we get,    $dy space equals space left parenthesis sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x right parenthesis space dx$
$therefore space space space space integral dy space equals space integral left parenthesis sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x right parenthesis space dx therefore space space integral dy space equals space integral sin cubed straight x space cos squared straight x space dx space plus space integral straight x space straight e to the power of straight x space dx space space space space space space... space left parenthesis 1 right parenthesis$
Let    $straight I subscript 1 space equals space integral sin cubed straight x space space cos squared straight x space dx space equals space integral sin squared straight x space cos squared straight x. space sinx space dx$
$equals space integral left parenthesis 1 minus cos squared straight x right parenthesis space cos squared straight x. space sinx space dx$
Put cosx =t, $therefore space space space space sin space straight x space dx space equals space minus dt$
$therefore space space space straight I subscript 1 space equals space minus integral left parenthesis 1 minus straight t squared right parenthesis space straight t squared space dt space equals space minus integral left parenthesis straight t squared minus straight t to the power of 4 right parenthesis space dt space equals space minus open parentheses straight t cubed over 3 minus straight t to the power of 5 over 5 close parentheses equals 1 fifth straight t to the power of 5 minus 1 third straight t cubed$
$equals space 1 fifth cos to the power of 5 straight x space minus space 1 third cos cubed straight x$

Let $straight I subscript 2 space equals space integral straight x space straight e to the power of straight x space dx space equals space straight x space. straight e to the power of straight x space minus space integral 1. space straight e to the power of straight x space dx space equals space straight x space straight e to the power of straight x space minus space straight e to the power of straight x space equals space left parenthesis straight x minus 1 right parenthesis space straight e to the power of straight x$
$therefore space space space from space left parenthesis 1 right parenthesis comma space space integral dy space equals space 1 fifth cos to the power of 5 straight x minus 1 third cos cubed straight x plus left parenthesis straight x minus 1 right parenthesis space straight e to the power of straight x therefore space space space straight y space equals space 1 fifth cos to the power of 5 straight x space minus space 1 third cos cubed straight x space plus space left parenthesis straight x minus 1 right parenthesis space straight e to the power of straight x plus straight c space is space the space required space solution. space$

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