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

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95.
Solve $straight x to the power of 5 dy over dx space equals space minus straight y to the power of 5$

The given differential equation is
$straight x to the power of 5 dy over dx equals negative straight y to the power of 5 space space space space or space space space space space dy over dx space equals space minus straight y to the power of 5 over straight x to the power of 5$
Separating the variables, we get,
$1 over straight y to the power of 5 dy space equals space minus 1 over straight x to the power of 5 dx$
Integrating, $integral 1 over straight y to the power of 5 dy space equals space minus integral 1 over straight x to the power of 5 dx space space or space space space integral straight y to the power of negative 5 end exponent dy space equals negative integral straight x to the power of negative 5 end exponent dx$
$therefore space space space space space space fraction numerator straight y to the power of negative 4 end exponent over denominator negative 4 end fraction space equals space minus fraction numerator straight x to the power of negative 4 end exponent over denominator negative 4 end fraction plus straight c apostrophe space space space space space or space space space fraction numerator 1 over denominator negative 4 straight y to the power of 4 end fraction space equals space fraction numerator 1 over denominator 4 straight x to the power of 4 end fraction plus straight c apostrophe$
or $1 over straight x to the power of 4 plus 1 over straight y to the power of 4 space equals space minus 4 straight c apostrophe$
or $straight x to the power of negative 4 end exponent plus straight y to the power of negative 4 end exponent space equals space straight c comma space space where space straight c space equals space minus 4 straight c apostrophe.$

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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: