Show that the following differential equation is homogeneous and

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 Multiple Choice QuestionsShort 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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 Multiple Choice QuestionsLong 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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 Multiple Choice QuestionsShort Answer Type

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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


The given differential equation is
                 xy apostrophe space minus space straight y plus space straight x space sin space open parentheses straight y over straight x close parentheses space equals space 0
therefore space space space space space space space space straight x dy over dx space equals space straight y minus straight x space sin straight y over straight x
therefore space space space space space space space space dy over dx space equals space straight y over straight x minus sin straight y over straight x
Put straight y space equals space straight v space straight x space so space that space space dy over dx space equals space straight v plus straight x dv over dx
therefore space space space space space space straight v plus straight x dv over dx space equals space straight v minus sin space straight v comma space space space space space space space space space space space space space space space space therefore space space space straight x space dv over dx space equals space minus sin space straight v

or            fraction numerator 1 over denominator sin space straight v end fraction dv space equals space minus 1 over straight x dx
          integral space cosec space straight v space dv space equals space minus integral 1 over straight x dx
therefore space space space space space space log space space open vertical bar tan space straight v over 2 close vertical bar space equals space minus log space open vertical bar straight x close vertical bar space plus space log space open vertical bar straight c close vertical bar
or       log space open vertical bar straight x space tan space straight v over 2 close vertical bar space equals space log space open vertical bar straight c close vertical bar space space space space space space space space space space space space or space space space space straight x space tan straight v over 2 space equals straight c
or     straight x space tan space open parentheses fraction numerator straight y over denominator 2 straight x end fraction close parentheses space equals space straight c space is space the space required space solution. space

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 Multiple Choice QuestionsLong 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
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