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Differential Equations

Question

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

Solution

The given differential equation is

straight y space dx space plus space left parenthesis straight x minus straight y squared right parenthesis space dy space equals space 0

or

straight y dx over dy plus straight x minus straight y squared space equals space 0 space space space space space or space space space space straight y dx over dy plus straight x space equals space straight y squared space space space or space space dx over dy plus 1 over straight y straight x space equals space straight y

Comparing it with

dx over dy plus Px space equals space straight Q comma space space we space get comma space space straight P space equals space 1 over straight y comma space space straight Q space equals space straight y

therefore space space space space integral straight P space dy space equals space integral 1 over straight y dy space equals log space straight y comma space space space straight e to the power of integral straight P space dy end exponent space equals space straight e to the power of log space straight y end exponent space equals space straight y

therefore space space

solution of differential equation is

straight x space straight e to the power of integral straight P space dy end exponent space equals space integral straight Q. space straight e to the power of integral straight P space dy end exponent space dy space plus space straight c space space or space space space straight x space straight y space equals space integral straight y. space straight y space dy space plus straight c

or

straight x space straight y space space equals space straight y cubed over 3 plus straight c

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