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Differential Equations

Short Answer Type

101. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:

straight e to the power of straight y dx plus straight e to the power of straight x dy space equals space 0

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102. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:

left parenthesis sinx plus cosx right parenthesis space dy space equals space left parenthesis cosx minus sinx right parenthesis space dx

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103. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:
$straight y apostrophe space equals left parenthesis 1 plus straight x squared right parenthesis thin space left parenthesis 1 plus straight y squared right parenthesis$

The given differential equation is

straight y apostrophe space equals left parenthesis 1 plus straight x squared right parenthesis thin space left parenthesis 1 plus straight y squared right parenthesis space space space space or space space space space dy over dx equals left parenthesis 1 plus straight x squared right parenthesis thin space left parenthesis 1 plus straight y squared right parenthesis

Separating the variables, we get,

fraction numerator 1 over denominator 1 plus straight y squared end fraction dy space equals space left parenthesis 1 plus straight x squared right parenthesis space dx

Integrating,

integral fraction numerator 1 over denominator 1 plus straight y squared end fraction dy space equals space integral left parenthesis 1 plus straight x squared right parenthesis space dx

therefore space space space space space space space tan to the power of negative 1 end exponent straight y space equals space straight x plus straight x cubed over 3 plus straight c comma space space straight x space element of space straight R space is space the space solution. space

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104. Find a one-parameter family of solutions of the following differential equation, indicating carefully the interval in which the solutions are valid:

left parenthesis straight e to the power of straight x plus straight e to the power of straight x right parenthesis space dy space minus space left parenthesis straight e to the power of straight x minus straight e to the power of straight x right parenthesis space dx space equals space 0

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105.

Solve the differential equation
$left parenthesis tan squared straight x plus 2 space tanx space plus 5 right parenthesis space dy over dx space equals space 2 left parenthesis 1 plus tanx right parenthesis space sec squared straight x.$