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Vector Algebra

Question
CBSEENMA12033061

Show that the general solution of the differential equation dy over dx plus fraction numerator straight y squared plus straight y plus 1 over denominator straight x squared plus straight x plus 1 end fraction space equals 0 is given by (x + y + 1) = A (1 – x – y – 2 x y), where A is parameter.

Solution

The given differential equation is
                      dy over dx plus fraction numerator straight y squared plus straight y plus 1 over denominator straight x squared plus straight x plus 1 end fraction space equals space 0 space or space space fraction numerator dy over denominator straight y squared plus straight y plus 1 end fraction plus fraction numerator dx over denominator straight x squared plus straight x plus 1 end fraction space equals space 1
Integrating, we get
                         integral fraction numerator dy over denominator straight y squared plus straight y plus 1 end fraction plus integral fraction numerator dx over denominator straight x squared plus straight x plus 1 end fraction space equals space straight C
rightwards double arrow space space space space space integral space fraction numerator dy over denominator open parentheses straight y plus begin display style 1 half end style close parentheses squared plus open parentheses 1 minus begin display style 1 fourth end style close parentheses end fraction plus integral fraction numerator dx over denominator open parentheses straight x plus begin display style 1 half end style close parentheses squared plus 1 minus begin display style 1 fourth end style end fraction space equals space straight C
rightwards double arrow space space space space integral space fraction numerator dy over denominator open parentheses straight y plus begin display style 1 half end style close parentheses squared plus open parentheses begin display style fraction numerator square root of 3 over denominator 2 end fraction end style close parentheses squared end fraction plus integral fraction numerator dx over denominator open parentheses straight x plus begin display style 1 half end style close parentheses squared plus open parentheses begin display style fraction numerator square root of 3 over denominator 2 end fraction end style close parentheses squared end fraction space equals space straight C
rightwards double arrow space space space space fraction numerator 2 over denominator square root of 3 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator straight y plus begin display style 1 half end style over denominator begin display style fraction numerator square root of 3 over denominator 2 end fraction end style end fraction close parentheses plus fraction numerator 2 over denominator square root of 3 end fraction tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus begin display style 1 half end style over denominator begin display style fraction numerator square root of 3 over denominator 2 end fraction end style end fraction close parentheses space equals space straight C
rightwards double arrow space space tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight y plus 1 over denominator square root of 3 end fraction close parentheses space plus space tan to the power of negative 1 end exponent space open parentheses fraction numerator 2 straight x plus 1 over denominator square root of 3 end fraction close parentheses space equals space fraction numerator square root of 3 straight C over denominator 2 end fraction space equals space straight A subscript 1 space left parenthesis say right parenthesis
Taking tangents on the two sides, we get
                     fraction numerator begin display style fraction numerator 2 straight y plus 1 over denominator square root of 3 end fraction end style plus begin display style fraction numerator 2 straight x plus 1 over denominator square root of 3 end fraction end style over denominator 1 minus open parentheses begin display style fraction numerator 2 straight y plus 1 over denominator square root of 3 end fraction end style close parentheses space open parentheses begin display style fraction numerator 2 straight x plus 1 over denominator square root of 3 end fraction end style close parentheses end fraction space equals space tan space straight A subscript 1 space space rightwards double arrow space space space fraction numerator 2 square root of 3 space left parenthesis straight x plus straight y plus 1 right parenthesis over denominator 3 minus left parenthesis 4 space straight x space straight y space plus 2 straight x plus 2 straight y plus 1 right parenthesis end fraction equals space tan space straight A subscript 1
rightwards double arrow space space space space space space fraction numerator 2 square root of 3 left parenthesis straight x plus straight y plus 1 right parenthesis over denominator 2 left parenthesis 1 minus straight x minus straight y minus 2 xy right parenthesis end fraction space equals space tan space straight A subscript 1
rightwards double arrow space space space space straight x plus straight y plus 1 space equals space fraction numerator 1 over denominator square root of 3 end fraction tan space straight A subscript 1 space left parenthesis 1 minus straight x minus straight y minus 2 xy right parenthesis
rightwards double arrow space space space straight x plus straight y plus 1 space equals space straight A left parenthesis 1 minus straight x minus straight y minus 2 xy right parenthesis comma
             where straight A space equals space fraction numerator 1 over denominator square root of 3 end fraction space space tan space straight A subscript 1 space is space an space arbitrary space constant. space

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