Differential Equations

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Find the general solution of the differential equation y dx – (x + 2 y 2 ) dy = 0. - Wired Faculty

Question

Find the general solution of the differential equation y dx – (x + 2 y²) dy = 0.

Solution

The given differential equation is
y dx – (x + 2 y²) dy = 0
or

straight y dx over dy minus straight x minus 2 straight y squared space equals space 0 space space space space or space space space straight y dx over dy minus straight x space equals space 2 straight y squared

dx over dy minus 1 over straight y straight x space equals space 2 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 minus 1 over straight y comma space space straight Q space equals space 2 straight y

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

Solution of differential equation is

xe 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 dy space plus space straight c

straight x. space 1 over straight y space equals space integral space 2 straight y. space 1 over straight y dy plus straight c

straight x over straight y space equals space 2 space integral space 1 space dy space plus straight c space space space or space space space straight x over straight y space equals space 2 straight y plus straight c

straight x equals space 2 straight y squared plus straight c space straight y space is space the space required space solution. space