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Differential Equations

Question

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Solve:
$open parentheses straight x plus 2 straight y cubed close parentheses space straight y apostrophe space equals space straight y comma space space straight y greater than 0$

Solution

The given equation is

left parenthesis straight x plus 2 straight y cubed right parenthesis space dy over dx space equals space straight y

or space space open parentheses straight x plus 2 straight y cubed close parentheses. space space fraction numerator 1 over denominator begin display style dx over dy end style end fraction space equals space straight y space space space space space space or space space space space straight x plus 2 straight y cubed space equals space straight y space dy over dx or space space straight x over straight y plus 2 straight y squared space equals space dx over dy space space space space space space space or space space space space space space space dy over dx minus 1 over straight y. space straight x space equals space 2 space straight y squared

Comparing it with

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

straight P space equals negative 1 over straight y comma space space straight Q space equals space 2 straight y squared integral straight P space dy space equals space minus integral 1 over straight y dy space equals space minus log space straight y

straight I. straight F. space equals space straight e to the power of integral space 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

therefore

solution of given equation is

straight x. space 1 over straight y space equals space integral space 2 straight y squared. space 1 over straight y dy plus straight c space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space straight x. space straight e to the power of integral straight P space dx end exponent space equals space integral straight Q. space straight e to the power of integral straight P space dx end exponent dy plus straight c close square brackets

or space space space space straight x over straight y space equals space 2 integral straight y space dy space plus straight c or space space space straight x over straight y space equals space straight y squared plus straight c comma space space or space space straight x space equals space straight y cubed plus straight c space straight y

which is required solution.

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