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Differential Equations

Question

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Solve: $straight x squared dy over dx space equals straight x squared plus 5 xy plus 4 straight y squared.$

Solution

The given differential equation is

straight x squared dy over dx space equals space straight x squared plus 5 xy plus 4 straight y squared

or

dy over dx space equals space fraction numerator straight x squared plus 5 xy plus 4 straight y squared over denominator straight x squared end fraction

Put y = v x so that

dy over dx equals straight v plus straight x dv over dx

therefore space space space space space from space left parenthesis 1 right parenthesis comma space space straight v plus straight x dv over dx space equals fraction numerator straight x squared plus 5 straight x. space straight v space straight x space plus space 4 space straight v squared space straight x squared over denominator straight x squared end fraction

or

straight v plus straight x dv over dx space equals space 1 plus 5 straight v plus 4 straight v squared

or

straight x dv over dx space equals space 4 space straight v squared plus 4 space straight v space plus space 1

Separating the variables, we get,

fraction numerator 1 over denominator 4 space straight v squared plus 4 space straight v space plus space 1 end fraction dv space equals space 1 over straight x dx

Integrating

integral fraction numerator 1 over denominator 4 straight v squared plus 4 straight v plus 1 end fraction straight d space straight v space equals space integral 1 over straight x dx

or

integral fraction numerator 1 over denominator left parenthesis 2 space straight v space plus space 1 right parenthesis squared end fraction dv space equals space integral 1 over straight x dx space space space space space or space space space space integral left parenthesis 2 straight v plus 1 right parenthesis to the power of negative 2 end exponent space dv space equals space integral 1 over straight x dx

therefore space space space space fraction numerator left parenthesis 2 space straight v space plus space 1 right parenthesis to the power of negative 1 end exponent over denominator left parenthesis 2 right parenthesis space left parenthesis negative 1 right parenthesis end fraction space equals space log space open vertical bar straight x close vertical bar space plus straight c space space space space space or space space space space minus fraction numerator 1 over denominator 2 space left parenthesis 2 space straight v space plus 1 right parenthesis end fraction log space open vertical bar straight x close vertical bar plus straight c

or

negative fraction numerator 1 over denominator 2 open parentheses 2 begin display style straight y over straight x end style plus 1 close parentheses end fraction space equals space log space open vertical bar straight x close vertical bar plus straight c space space space space space space space space or space space space space space space space space space minus fraction numerator straight x over denominator 2 space left parenthesis straight x plus 2 space straight y right parenthesis end fraction space equals space log space open vertical bar straight x close vertical bar plus straight c

or

fraction numerator straight x over denominator 2 space left parenthesis straight x plus 2 space straight y right parenthesis end fraction plus log space open vertical bar straight x close vertical bar plus straight c space equals space 0

is the required solution.

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