Which of the following is true for

Question:

Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{d y}{d x}=x y-1+x-y ; y(0)=0$

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

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

  3. (3) $y(1)=1$

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


Correct Option: 1

Solution:

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

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

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

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

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