Show that the given differential equation is homogeneous and sol

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 Multiple Choice QuestionsShort Answer Type

141.

Solve:
open parentheses straight e to the power of straight x plus 1 close parentheses space straight y space dy space plus space left parenthesis straight y plus 1 right parenthesis space straight e to the power of straight x space dx space equals space 0


 

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142.

Solve:
straight y left parenthesis 1 minus straight x squared right parenthesis space dy space plus space straight x space left parenthesis 1 plus straight y squared right parenthesis space dx space equals space 0.



 

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143.

Solve:
straight y space logydx space minus space straight x space dy space equals space 0



 

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 Multiple Choice QuestionsLong Answer Type

144.

Show that the general solution of the differential equation dy over dx plus fraction numerator straight y squared plus straight y plus 1 over denominator straight x squared plus straight x plus 1 end fraction space equals 0 is given by (x + y + 1) = A (1 – x – y – 2 x y), where A is parameter.

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 Multiple Choice QuestionsShort Answer Type

145.

Solve:
dy over dx space equals space fraction numerator straight x left parenthesis 2 space log space straight x space plus space 1 right parenthesis over denominator sin space straight y space plus space straight y space cosy end fraction

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146.

Solve
dy over dx space equals space fraction numerator xe to the power of straight x logx plus straight e to the power of straight x over denominator straight x space cosy end fraction

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 Multiple Choice QuestionsLong Answer Type

147.

Solve
dy over dx space equals space sin cubed straight x space cos squared straight x plus straight x space straight e to the power of straight x

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148.

Solve:
dy over dx space equals space cos cubed straight x space sin to the power of 4 straight x plus straight x square root of 2 straight x plus 1 end root


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149.

Solve:
dy over dx equals negative straight x space sin squared straight x space space equals space fraction numerator 1 over denominator straight x space log space straight x end fraction



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150.

Show that the given differential equation is homogeneous and solve it.
(x2 – y2) dx + 2xy dy = 0
given that y = 1 when x = 1.


The given differential equation is
    open parentheses straight x squared minus straight y squared close parentheses space dx space plus space 2 xy space dy space equals space 0 space space space space or space space space 2 xy space dy space equals space left parenthesis straight y squared minus straight x squared right parenthesis space dx
or             dy over dx space equals space fraction numerator straight y squared minus straight x squared over denominator 2 xy end fraction
Put y = vx so that dy over dx equals space straight v plus straight x dv over dx

therefore space space space space straight v plus straight x dv over dx equals fraction numerator straight v squared straight x squared minus straight x squared over denominator 2 vx squared end fraction space space space or space space space straight v plus straight x dv over dx space equals space fraction numerator straight v squared minus 1 over denominator 2 space straight v end fraction

therefore space space space space space space space space space space space straight x dv over dx equals fraction numerator straight v squared minus 1 over denominator 2 space straight v end fraction minus straight v space space space or space space space straight x dv over dx space equals space fraction numerator straight v squared minus 1 minus 2 straight v squared over denominator 2 straight v end fraction
therefore space space space space space space space space straight x dv over dx space equals space fraction numerator negative 1 minus straight v squared over denominator 2 space straight v end fraction space space space space space rightwards double arrow space space space space space space space fraction numerator 2 space straight v over denominator 1 plus straight v squared end fraction dv space equals space minus 1 over straight x dx
therefore space space space space space space integral fraction numerator 2 space straight v over denominator 1 plus straight v squared end fraction dv space equals space minus integral 1 over straight x dx
therefore space space space log space open vertical bar 1 plus straight v squared close vertical bar space equals space minus log space open vertical bar straight x close vertical bar plus straight c apostrophe
therefore space space space log space open vertical bar 1 plus straight v squared close vertical bar plus log space open vertical bar straight x close vertical bar space equals space straight c apostrophe
therefore space log space open vertical bar left parenthesis 1 plus straight v squared right parenthesis space left parenthesis straight x right parenthesis close vertical bar space equals space straight c apostrophe
therefore space space space space space space straight x left parenthesis 1 plus straight v squared right parenthesis space equals space straight c apostrophe space space space space space space space rightwards double arrow space space space space space straight x space open parentheses 1 plus straight y squared over straight x squared close parentheses space equals space straight c
therefore space space space space space space straight x squared plus straight y squared space equals space straight c space straight x
Now,  straight y space equals space 1 comma space space when space straight x space equals space 1

therefore space space space space space space space space space space 1 plus 1 space equals space straight c space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight c space equals space 2
therefore space space space solution space is space straight x squared plus straight y squared space equals space 2 straight x.

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