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

The given differential equation is

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

therefore space space space space space space straight e to the power of straight y dy space equals space minus straight e to the power of straight x dx

Integration,

integral straight e to the power of negative straight y end exponent space dy space equals negative integral straight e to the power of negative straight x end exponent dx

therefore space space space space space fraction numerator straight e to the power of negative straight y end exponent over denominator negative 1 end fraction space equals negative fraction numerator straight e to the power of negative straight x end exponent over denominator negative 1 end fraction plus straight c subscript 1 space or space space straight e to the power of straight y plus straight e to the power of straight x space equals space straight c comma space space straight x space element of space straight R

is the solution.

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