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Differential Equations

Question

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Solve the following differential equation:
$open parentheses straight x plus straight y close parentheses squared dy over dx space equals straight a squared$

Solution

The given differential equation is

left parenthesis straight x plus straight y right parenthesis squared space dy over dx space equals space straight a squared

Put

straight x plus straight y space equals straight t comma space space space space space therefore space space 1 plus dy over dx space space equals space dt over dx space space or space space dy over dx equals space dt over dx minus 1

therefore space space space space from space left parenthesis 1 right parenthesis comma space space straight t squared space open parentheses dt over dx minus 1 close parentheses space equals straight a squared space or space space straight t squared space dt over dx space equals space straight a squared plus straight t squared space space or space space space fraction numerator straight t squared over denominator straight a squared plus straight t squared end fraction dt space equals space dx

or

integral fraction numerator straight t squared over denominator straight a squared plus straight t squared end fraction dt space equals space integral dx

or

integral open parentheses 1 minus fraction numerator straight a squared over denominator straight a squared plus straight t squared end fraction close parentheses dt space equals space straight x space space or space straight t minus straight a squared space 1 over straight a tan to the power of negative 1 end exponent straight t over straight a space equals straight x plus straight c

or

straight x plus straight y space minus straight a space tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus straight y over denominator straight a end fraction close parentheses space equals space straight x plus straight c

or

straight y minus straight a space tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus straight y over denominator straight a end fraction close parentheses space equals space straight c space is space the space required space solution. space

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