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Continuity And Differentiability

Question
CBSEENMA12036017
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If a curve y=f(x) passes through the point (1, −1) and satisfies the differential equation, y(1+xy) dx=x dy, then f(-1/2) is equal to

  • -2/5

  • -4/5

  • 2/5

  • 4/5

Solution

D.

4/5

Given differential equation is
 y( 1+ xy) dx = xdy
⇒ ydx + xy2 dx = xdy
fraction numerator xdy minus ydx over denominator straight y squared end fraction space equals space xdx rightwards double arrow space minus fraction numerator left parenthesis ydx minus xdy right parenthesis over denominator straight y squared end fraction space equals space xdx On space integrating space both space sides comma space we space get fraction numerator negative straight x over denominator straight y end fraction space equals space straight x squared over 2 plus straight C therefore comma space it space passes space through space left parenthesis 1 comma negative 1 right parenthesis 1 space equals 1 half space plus straight C rightwards double arrow straight C space equals space 1 half Now comma space from space eq space left parenthesis straight i right parenthesis comma minus straight x over straight y space equals space straight x squared over 2 space plus space 1 half rightwards double arrow space straight x squared space plus space 1 space equals space minus fraction numerator 2 straight x over denominator straight y end fraction straight y equals space minus space fraction numerator 2 straight x over denominator straight x squared plus 1 end fraction therefore comma space straight f open parentheses negative fraction numerator begin display style 1 end style over denominator 2 end fraction close parentheses space equals space 4 over 5

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