Which of the following is true for $y\left(x\right)$ that satisfies the differential equation

$\frac{dy}{dx}=xy-1+x-y; y\left(0\right)=0$

• Correct answer
  A

  $y\left(1\right)={\mathrm{e}}^{-\frac{1}{2}}-1$
• B

  $y\left(1\right)=e^{\frac{1}{2}}-e^{-\frac{1}{2}}$
• C

  $y\left(1\right)=1$
• D

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

Hence, the correct answer is option: 'A'.