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Differential Equations

Short Answer Type

191.

Solve the differential equation:
$straight x squared dy over dx space equals straight y left parenthesis straight x plus straight y right parenthesis.$

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Long Answer Type

192.

Solve the differential equation:
$left parenthesis straight y plus straight x right parenthesis space dy over dx space equals space straight y minus straight x.$

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193.

Solve the following differential equation:
(y² – x²) dy = 3 x y dx

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194. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:
(x - y) y' = x + 2 y

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195. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:
(x² + y²) y' = 8 x² - 3 x y + 2 y²

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196. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:
(3 x y + y²) dx = (x² + x y) dy

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197. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:
2 x y dx + (x² + 2 y²) dy = 0

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198. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:

left parenthesis 2 straight x squared straight y plus straight y cubed right parenthesis space dx plus space left parenthesis xy squared minus 3 straight x cubed right parenthesis space dy space equals space 0

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Short Answer Type

199. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:

straight x space straight y apostrophe space minus space straight y space plus space straight x space sin space space open parentheses straight y over straight x close parentheses space equals 0

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Long Answer Type

200. Show that the following differential equation is homogeneous and find a primitive of it. Derive the solution wherever possible:
$left parenthesis straight x plus 2 straight y right parenthesis space dx space minus space left parenthesis 2 straight x minus straight y right parenthesis space dy space equals space 0$

The given differential equation is
   $left parenthesis straight x plus 2 straight y right parenthesis space dx space minus space left parenthesis 2 straight x minus straight y right parenthesis space dy space equals space 0$
or    $dy over dx space equals space fraction numerator straight x plus 2 straight y over denominator 2 straight x minus straight y end fraction$

Put y = v x so that    $dy over dx space equals space straight v plus straight x dv over dx$

$therefore space space space straight v plus straight x dv over dx space equals fraction numerator straight x plus 2 vx over denominator 2 straight x minus vx end fraction therefore space space space space straight v plus straight x dv over dx space equals space fraction numerator 1 plus 2 straight v over denominator 2 minus straight v end fraction therefore space space space space space straight x dv over dx space equals space fraction numerator 1 plus 2 straight v over denominator 2 minus straight v end fraction minus straight v therefore space space space space straight x dv over dx space equals space fraction numerator 1 plus 2 straight v minus 2 straight v plus straight v squared over denominator 2 minus straight v end fraction therefore space space space space space space straight x dv over dx space equals space fraction numerator 1 plus straight v squared over denominator 2 minus straight v end fraction therefore space space space fraction numerator 2 minus straight v over denominator 1 plus straight v squared end fraction dv space equals space 1 over straight x dx therefore space space space space integral fraction numerator 2 minus straight v over denominator 1 plus straight v squared end fraction dv space equals space integral 1 over straight x dx therefore space space space 2 space integral fraction numerator 1 over denominator 1 plus straight v squared end fraction dv minus 1 half integral fraction numerator 2 straight v over denominator 1 plus straight v squared end fraction dv space equals space integral 1 over straight x dx therefore space 2 space tan to the power of negative 1 end exponent straight v space minus space 1 half log space left parenthesis 1 plus straight v squared right parenthesis space equals space log space open vertical bar straight x close vertical bar plus space log space straight c subscript 1 therefore space space 4 space tan to the power of negative 1 end exponent straight v space minus space log space left parenthesis 1 plus straight v squared right parenthesis space equals space 2 space log space open vertical bar straight x close vertical bar space plus space 2 space log space straight c subscript 1 therefore space space 4 space tan to the power of negative 1 end exponent straight y over straight x minus log space open parentheses 1 plus straight y squared over straight x squared close parentheses space equals space log space straight x squared plus space log space straight c therefore space space space 4 space tan to the power of negative 1 end exponent straight y over straight x minus log space left parenthesis straight x squared plus straight y squared right parenthesis space plus space log space straight x squared space equals space log space straight x squared plus space log space straight c therefore space space space 4 space tan to the power of negative 1 end exponent straight y over straight x minus log space left parenthesis straight x squared plus straight y squared right parenthesis space equals space log space straight c$
which is the required solution.

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