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Which of the following is true for y(x) that satisfies the differential equation $${{dy} \over {dx}}$$ = xy $$-$$ 1 + x $$-$$ y; y(0) = 0 :

JEE · Math · previous-year question

  1. A.y(1) = 1
  2. B.y(1) = e$$-$$$${1 \over 2}$$ $$-$$ 1correct
  3. C.y(1) = e$${1 \over 2}$$ $$-$$ e$$-$$$${1 \over 2}$$
  4. D.y(1) = e$${1 \over 2}$$ $$-$$ 1

Answer

B. y(1) = e$$-$$$${1 \over 2}$$ $$-$$ 1

Explanation

$${{dy} \over {dx}} = (x - 1)y + (x - 1)$$ $${{dy} \over {dx}} = (x - 1)(y + 1)$$ $${{dy} \over {y + 1}} = (x - 1)dx$$ Integrating both sides, we get $$\ln (y + 1) = {{{x^2}} \over 2} - x + c$$ $$x = 0,y = 0$$ $$ \Rightarrow c = 0$$ $$ \therefore $$ $$\ln (y + 1) = {{{x^2}} \over 2} - x$$ putting $$x = 1,\ln (y + 1) = {1 \over 2} - 1 = - {1 \over 2}$$ $$y + 1 = {e^{ - {1 \over 2}}}$$ $$y = {e^{ - {1 \over 2}}} - 1$$ $$ \therefore $$ $$y(1) = {e^{ - {1 \over 2}}} - 1$$

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