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Differential Equations

Long Answer Type

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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Short Answer Type

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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Long Answer Type

208. Show that the given differential equation is homogeneous and solve it:
$straight x squared dy over dx space equals space straight x squared minus 2 straight y squared plus straight x space straight y$

The given differential equation is

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

dy over dx space equals fraction numerator straight x squared minus 2 straight y squared plus straight x space straight y over denominator straight x squared end fraction

...(1)
It is a differential equation of the form

dy over dx space equals space straight F left parenthesis straight x comma space straight y right parenthesis

Here,

straight F left parenthesis straight x comma space straight y right parenthesis space equals space fraction numerator straight x squared minus 2 straight y squared plus straight x space straight y over denominator straight x squared end fraction

Replacing x by

λx

and y by

λy

, we get.

straight F left parenthesis λx comma space λy right parenthesis space equals space fraction numerator straight lambda squared straight x squared minus 2 straight lambda squared straight y squared plus straight lambda squared straight x space straight y over denominator straight lambda squared straight x squared end fraction space equals space fraction numerator straight lambda squared left parenthesis straight x squared minus 2 straight y squared plus xy right parenthesis over denominator straight lambda squared straight x squared end fraction space equals space straight lambda degree space left square bracket straight F left parenthesis straight x comma space straight y right parenthesis right square bracket

∴ F(x, y) is a homogeneous function of degree zero.
∴ given differential equation is a homogeneous differential equation.
Put y = vx $rightwards double arrow space space dy over dx space equals space straight v. space 1 space space plus straight x space dv over dx$
Substituting these values of y and $dy over dx$ in the given equation, we get
$straight v plus straight x dv over dx space equals space fraction numerator straight x squared minus 2 straight v squared straight x squared plus vx squared over denominator straight x squared end fraction$
or $straight v plus straight x dv over dx equals 1 minus 2 straight v squared plus straight v space space space space space rightwards double arrow space space space space space straight x dv over dx equals 1 minus 2 straight v squared$
$rightwards double arrow space space space space space space space space 1 over straight x dx space equals space fraction numerator 1 over denominator 1 minus 2 straight v squared end fraction dv$
Integrating, $integral 1 over straight x dx space equals space integral fraction numerator 1 over denominator 1 minus 2 straight v squared end fraction dv$
$therefore space space space space space integral 1 over straight x dx space equals space 1 half integral fraction numerator dv over denominator open parentheses begin display style fraction numerator 1 over denominator square root of 2 end fraction end style close parentheses squared minus straight v squared end fraction therefore space space space space space log space open vertical bar straight x close vertical bar space space equals 1 half. space fraction numerator 1 over denominator 2. space begin display style fraction numerator 1 over denominator square root of 2 end fraction end style end fraction log space open vertical bar fraction numerator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style plus straight v over denominator begin display style fraction numerator 1 over denominator square root of 2 end fraction end style minus straight v end fraction close vertical bar plus straight c rightwards double arrow space space space space space space space log space open vertical bar straight x close vertical bar space equals space fraction numerator 1 over denominator 2 square root of 2 end fraction space log space space open vertical bar fraction numerator 1 plus square root of 2 space straight v over denominator 1 minus square root of 2 straight v end fraction close vertical bar plus straight c rightwards double arrow space space space log space open vertical bar straight x close vertical bar space equals space fraction numerator 1 over denominator 2 square root of 2 end fraction log space open vertical bar fraction numerator 1 plus square root of 2. space begin display style straight y over straight x end style over denominator 1 minus square root of 2. begin display style straight y over straight x end style end fraction close vertical bar plus straight c rightwards double arrow space space space space log space open vertical bar straight x close vertical bar space equals space fraction numerator 1 over denominator 2 square root of 2 end fraction log space open vertical bar fraction numerator straight x plus square root of 2 straight y over denominator straight x minus square root of 2 straight y end fraction close vertical bar plus straight c$