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Differential Equations

Short Answer Type

121. Find a one parameter family of solutions of each of the following differential equation:
(x y² + 2x) dx + (x² y + 2y) dy = 0

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122. Find a one parameter family of solutions of each of the following differential equation:
x y y' = 1 + x + y + x y

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123.

Solve:
$dy over dx equals straight e to the power of straight x minus straight y end exponent plus straight x squared straight e to the power of straight y$

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124.

Solve:
$dy over dx space equals straight e to the power of straight x plus straight y end exponent plus straight x squared space straight e to the power of straight y.$

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125.

Solve:
$dy over dx space equals straight e to the power of straight x minus straight y end exponent plus straight x cubed straight e to the power of negative straight y end exponent$

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126.

Solve:
x (1 + y²) dx + y (1 + x²)dy = 0

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127.
Solve:
(1 + x) (1 + y²) dx + (1 + y) (1 + x²) dy = 0.

The given differential equation is
(1 + x) (1 + y²) dx + (1 + y) (1 + x²) dy = 0
or

left parenthesis 1 plus straight x right parenthesis space left parenthesis 1 plus straight y squared right parenthesis space dx space equals space minus left parenthesis 1 plus straight y right parenthesis thin space left parenthesis 1 plus straight x squared right parenthesis space dy

fraction numerator 1 plus straight y over denominator 1 plus straight y squared end fraction dy space equals space minus fraction numerator 1 plus straight x over denominator 1 plus straight x squared end fraction dx space space space rightwards double arrow space space space integral fraction numerator 1 plus straight y over denominator 1 plus straight y squared end fraction dy space equals space minus integral fraction numerator 1 plus straight x over denominator 1 plus straight x squared end fraction dx

therefore space space space space integral fraction numerator 1 over denominator 1 plus straight y squared end fraction dy plus 1 half integral fraction numerator 2 straight y over denominator 1 plus straight y squared end fraction dy space equals space minus integral fraction numerator 1 over denominator 1 plus straight x squared end fraction dx minus 1 half integral fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction dx therefore space space space space space tan to the power of negative 1 end exponent straight y space plus space 1 half space log space left parenthesis 1 plus straight y squared right parenthesis space equals space minus tan to the power of negative 1 end exponent straight x space minus space 1 half log left parenthesis 1 plus straight x squared right parenthesis space plus straight c

tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent straight y plus 1 half open square brackets log space left parenthesis 1 plus straight x squared right parenthesis space plus space log space left parenthesis 1 plus straight y squared right parenthesis close square brackets space equals space straight c

tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent straight y plus 1 half log space open square brackets left parenthesis 1 plus straight x squared right parenthesis thin space left parenthesis 1 plus straight y squared right parenthesis close square brackets space equals space straight c space is space the space required space solution. space

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128.

For the following differential equation, find the general solution:
$dy over dx equals fraction numerator 1 minus cosx over denominator 1 plus cosx end fraction$