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

1 over straight x cos straight y over straight x dx minus open parentheses straight x over straight y sin straight y over straight x plus cos straight y over straight x close parentheses dy space equals 0

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

2 space straight y space straight e to the power of straight x over 4 end exponent space dx plus open parentheses straight y minus 2 space straight x space straight e to the power of straight x over straight y end exponent close parentheses space dy space equals space 0

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

straight y space dx space plus space straight x space open parentheses log space straight y over straight x close parentheses space dy space minus space 2 space straight x space dy space equals space 0

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204. Show that the given differential equation is homogeneous and solve it:

left parenthesis straight x squared plus xy right parenthesis space dy space equals space left parenthesis straight x squared plus straight y squared right parenthesis space dx

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205. Show that the given differential equation is homogeneous and solve it:

straight y apostrophe space equals space fraction numerator straight x plus straight y over denominator straight x end fraction

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206. Show that the given differential equation is homogeneous and solve it:
(x-y) dy - (x+y) dx = 0

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207. Show that the given differential equation is homogeneous and solve it:

open parentheses straight x squared minus straight y squared close parentheses space dx space plus space 2 xy space dy space equals space 0

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208. Show that the given differential equation is homogeneous and solve it:

straight x squared dy over dx space equals space straight x squared minus 2 straight y squared plus straight x space straight y

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209. Show that the given differential equation is homogeneous and solve it:

open curly brackets straight x space cos space open parentheses straight y over straight x close parentheses plus straight y space sin space open parentheses straight y over straight x close parentheses close curly brackets straight y space dx space equals space open curly brackets straight y space sin space open parentheses straight y over straight x close parentheses minus straight x space cos space open parentheses straight y over straight x close parentheses close curly brackets space straight x space dy

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The given differential equation is

straight x space dy minus space straight y space dx space equals space square root of straight x squared plus straight y squared end root space dx

straight x space dy space equals space open parentheses straight y plus square root of straight x squared plus straight y squared end root close parentheses dx

dy over dx space equals space fraction numerator straight y plus square root of straight x squared plus straight y squared end root over denominator straight x 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 space left parenthesis 1 right parenthesis space becomes comma space space space space space straight v plus straight x dv over dx space equals space fraction numerator vx plus square root of straight x squared plus straight v squared straight x squared end root over denominator straight x end fraction

straight x dv over dx space equals straight v plus square root of 1 plus straight v squared end root minus straight v comma space space space or space space space space straight x dv over dx space equals space square root of 1 plus straight v squared end root

Separating the variables,

fraction numerator 1 over denominator square root of 1 plus straight v squared end root end fraction dv space equals space 1 over straight x dx

Integrating

integral fraction numerator 1 over denominator square root of 1 plus straight v squared end root end fraction dv space equals space integral 1 over straight x dx space space space space space space space space space space space space space space space space rightwards double arrow space space space space log space open vertical bar straight v plus square root of 1 plus straight v squared end root close vertical bar space equals space log space open vertical bar straight x close vertical bar space plus space log space straight c

open vertical bar straight v plus square root of 1 plus straight v squared end root close vertical bar space equals space straight c open vertical bar straight x close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space or space space space space space open vertical bar straight y over straight x plus square root of 1 plus straight y squared over straight x squared end root close vertical bar space equals space straight c open vertical bar straight x close vertical bar

open vertical bar straight y plus square root of straight x squared plus straight y squared end root space close vertical bar space equals space straight c open vertical bar straight x close vertical bar squared comma space space or space space space space open vertical bar straight y plus square root of straight x squared plus straight y squared end root close vertical bar space equals space straight c space straight x squared

therefore space space straight y plus square root of straight x squared plus straight y squared end root space equals space plus-or-minus space straight c space straight x squared space space space or space space space straight y plus square root of straight x squared plus straight y squared end root space equals space straight A space straight x squared space space where space straight A space equals space plus-or-minus straight c

which is required solution.

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