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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

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

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106. Solve the following differential equation:
x cos y dy = (x e^x log x + e^x) dx.

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107. Solve the following differential equation:

dy over dx space equals space log space left parenthesis straight x plus 1 right parenthesis

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108. Solve the following differential equation:
$left parenthesis straight y plus straight x space straight y right parenthesis space dx space plus space left parenthesis straight x minus straight x space straight y squared right parenthesis space dy space equals space 0$

The given differential equation is
$left parenthesis straight y plus straight x space straight y right parenthesis space dx space plus space left parenthesis straight x minus straight x space straight y squared right parenthesis space dy space equals space 0$
$or space straight y left parenthesis 1 plus straight x right parenthesis space dx space plus space straight x left parenthesis 1 minus straight y squared right parenthesis space dy space equals space 0 or space space straight x left parenthesis 1 minus straight y squared right parenthesis space dy space equals space minus straight y left parenthesis 1 plus straight x right parenthesis space dx or space fraction numerator 1 minus straight y squared over denominator straight y end fraction dy space equals space minus fraction numerator 1 plus straight x over denominator straight x end fraction dx$
Integrating, $integral fraction numerator 1 minus straight y squared over denominator straight y end fraction dy space equals space minus fraction numerator 1 plus straight x over denominator straight x end fraction dx$
$therefore space space space integral open parentheses 1 over straight y minus straight y close parentheses space dy space equals space minus integral open parentheses 1 over straight x plus 1 close parentheses space dx therefore space space log space open vertical bar straight y close vertical bar minus straight y squared over 2 space equals space minus log space straight x space minus space straight x space plus straight c comma space which space is space the space required space solution. space$