Which of the following


Which of the following is true for $\mathrm{y}(\mathrm{x})$ that satisfies the differential equation

$\frac{d y}{d x}=x y-1+x-y ; y(0)=0$

  1. $y(1)=e^{\frac{1}{2}}-1$

  2. $y(1)=e^{\frac{1}{2}}-e^{-\frac{1}{2}}$

  3. $y(1)=1$

  4. $y(1)=e^{\frac{1}{2}}-1$

Correct Option: 1


$\frac{d y}{d x}=(1+y)(x-1)$

$\frac{d y}{(y+1)}=(x-1) d x$

Integrate $\ln (y+1)=\frac{x^{2}}{2}-x+c$

$(0,0) \Rightarrow c=0 \Rightarrow y=e^{\left(\frac{x^{2}}{2}-x\right)}-1$

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