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Differential Equations

Multiple Choice Questions

91.

Which of the following differential equations has y = c₁e^x + c₂e^–x as the general solution?

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92. Which of the following differential equations has y = x as one of its particular solution?

$fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x squared dy over dx plus straight x space straight y space equals straight x$
$fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx plus straight x space straight y space equals space straight x$
$fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x squared dy over dx plus straight x space straight y space equals space 0$
$fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x squared dy over dx plus straight x space straight y space equals space 0$

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Short Answer Type

93.

Solve $straight y apostrophe minus dy over dx space equals space minus straight x over straight y space left parenthesis straight y not equal to 0 right parenthesis.$

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94.
Solve $dy over dx plus straight y space equals space 1 comma space space space straight y not equal to 1$

The given differential equation is

dy over dx plus straight y space equals space 1 space space space or space space space dy over dx equals space 1 minus straight y

Separating the variables, we get,

fraction numerator 1 over denominator 1 minus straight y end fraction dy space equals space dx

Integrating,

integral fraction numerator 1 over denominator 1 minus straight y end fraction dy space equals space integral 1 space dx

therefore space space space space space space fraction numerator log open vertical bar 1 minus straight y close vertical bar over denominator negative 1 end fraction space equals space straight x plus straight c apostrophe

therefore space space space space space log space open vertical bar 1 minus straight y close vertical bar space equals space minus straight x minus straight c apostrophe

therefore space space space space space space space space open vertical bar 1 minus straight y close vertical bar space equals space straight e to the power of negative straight x minus straight c apostrophe end exponent space space space or space space space open vertical bar 1 minus straight y close vertical bar space equals space straight e to the power of negative straight x end exponent. space straight e to the power of negative straight c apostrophe end exponent therefore space space space space space space space space space space space 1 minus straight y space equals space plus-or-minus straight e to the power of negative straight e apostrophe end exponent space straight e to the power of negative straight x end exponent therefore space space space space space space space space space space space space space 1 minus straight y space equals space minus ce to the power of negative straight x end exponent space space space space space space space space space space space space space space space space space space space space space space space where space minus straight c space equals space plus-or-minus straight e to the power of negative straight x end exponent therefore space space space space space space space space space space space straight y space equals space 1 plus ce to the power of negative straight x end exponent space space is space the space required space solution. space

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95.

Solve $straight x to the power of 5 dy over dx space equals space minus straight y to the power of 5$

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96.

Solve $dy over dx space equals fraction numerator 3 straight x squared over denominator 1 plus straight y squared end fraction$

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

dy over dx equals fraction numerator straight x plus 1 over denominator 2 minus straight y end fraction comma space space space space straight y space not equal to 2

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

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

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

dy over dx equals square root of 4 minus straight x squared end root comma space space minus 2 less than straight y less than 2

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