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Differential Equations

Question
CBSEENMA12033091
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Solve left parenthesis straight x minus straight y right parenthesis squared space dy over dx space equals space straight a squared.

Solution
The given differential equation is
                  left parenthesis straight x minus straight y right parenthesis squared dy over dx space equals space straight a squared                             ...(1)
Put x - y = t,       therefore space space space space 1 space minus space dy over dx space equals space dt over dx space space or space space dy over dx equals space 1 minus dt over dx
therefore space space space from space left parenthesis 1 right parenthesis comma space space space straight t squared space open parentheses 1 minus dt over dx close parentheses space equals space straight a squared space space space or space space space 1 minus dt over dx space equals space straight a squared over straight t squared
therefore space space space space space space space space space space space space space space space space space minus dt over dx space equals space straight a squared over straight t squared minus 1 space space space space space or space space space space dt over dx space equals space minus fraction numerator straight a squared minus straight t squared over denominator straight t squared end fraction
Separating the variables, we get,
               fraction numerator straight t squared over denominator straight a squared minus straight t squared end fraction dt space equals space minus dx
                                      negative straight t squared plus straight a squared space fraction numerator 1 over denominator long division enclose straight t squared end enclose end fraction space space space space space space space space space space space space space space space space space space straight t squared minus straight a squared space space space space space space space space space space space space space space space space space space space minus space space plus 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 space space space space space space space space straight a squared
Integrating,   integral fraction numerator straight t squared over denominator straight a squared minus straight t squared end fraction dt space equals space minus integral space 1 space dx
therefore space space space integral open parentheses negative 1 plus fraction numerator straight a squared over denominator straight a squared minus straight t squared end fraction close parentheses dt space equals space minus integral 1 space dx therefore space space space minus straight t plus straight a to the power of 2 space end exponent fraction numerator 1 over denominator 2 straight a end fraction space log space open vertical bar fraction numerator straight a plus straight t over denominator straight a minus straight t end fraction close vertical bar space equals space minus straight x plus straight c therefore space space minus straight t plus straight a over 2 log space open vertical bar fraction numerator straight a plus straight x minus straight y over denominator straight a minus straight x plus straight y end fraction close vertical bar space equals space minus straight x plus straight c therefore space space straight y minus straight x minus straight a over 2 log space open vertical bar fraction numerator straight a plus straight x minus straight y over denominator straight a minus straight x plus straight y end fraction close vertical bar space equals space minus straight x plus straight c therefore space space straight y minus straight x minus straight a over 2 log space open vertical bar fraction numerator straight a plus straight x minus straight y over denominator straight a minus straight x plus straight y end fraction close vertical bar space equals space minus straight x plus straight c
or   straight y minus straight a over 2 log space open vertical bar fraction numerator straight a plus straight x minus straight y over denominator straight a minus straight x plus straight y end fraction close vertical bar space equals space straight c comma which is required solution.

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