Find the particulars solution of the differential equation given that y =1 when x = 1. - Wired Faculty

Question

Find the particulars solution of the differential equation $straight x squared dy space equals space left parenthesis 2 xy plus straight y squared right parenthesis dx comma space$ given that y =1 when x = 1.

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Solution

straight x squared dy space equals space left parenthesis 2 xy plus straight y squared right parenthesis dx space space space rightwards double arrow dy over dx space equals space fraction numerator 2 xy plus straight y squared over denominator straight x squared end fraction space.... left parenthesis straight i right parenthesis space space space Let space straight y space equals space vx comma space space space space dy over dx equals straight v plus straight x dv over dx space space space Substituting space in space left parenthesis straight i right parenthesis comma space we space straight g space space space space space straight v plus straight x dv over dx space equals space fraction numerator 2 vx squared plus straight v squared straight x squared over denominator straight x squared end fraction space space rightwards double arrow space space straight v plus straight x dv over dx space equals space 2 straight v plus straight v squared space space space rightwards double arrow space straight x dv over dx equals space straight v squared plus straight v space space space rightwards double arrow space fraction numerator dv over denominator straight v squared plus straight v end fraction equals dx over straight x

Integrating both sides,

rightwards double arrow space integral fraction numerator dv over denominator straight v squared plus straight v end fraction equals integral dx over straight x rightwards double arrow integral fraction numerator straight v plus 1 minus straight v over denominator straight v left parenthesis straight v plus 1 right parenthesis end fraction. dv space equals space integral dx over straight x rightwards double arrow integral dv over straight v minus integral fraction numerator dv over denominator straight v plus 1 end fraction equals integral dx over straight x rightwards double arrow space logv minus log open vertical bar straight v plus 1 close vertical bar equals logx space plus space logC rightwards double arrow log open vertical bar fraction numerator straight v over denominator straight v plus 1 end fraction close vertical bar space equals space log open vertical bar Cx close vertical bar rightwards double arrow log open vertical bar fraction numerator begin display style bevelled straight y over straight x end style over denominator begin display style bevelled straight y over straight x end style plus 1 end fraction close vertical bar equals log open vertical bar Cx close vertical bar

rightwards double arrow space fraction numerator straight y over denominator straight y plus straight x end fraction equals Cx space left square bracket Removing space logarithm space in space both space sides right square bracket therefore space straight y equals Cxy plus Cx squared comma space which space is space the space general space solution. space Putting space straight y space equals space 1 space space and space straight x space equals space 1 comma 1 space equals space straight C space plus space straight C space space rightwards double arrow space space 2 straight C space equals space 1 space space space rightwards double arrow space straight C equals 1 half space space space straight y space equals space xy over 2 plus straight x squared over 2 space therefore 2 straight y space equals space xy plus straight y squared comma space which space is space the space particular space solution.

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