Welcome to Bytelearn!

Let’s check out your problem:

Lara tried to solve the differential equation $\frac{d y}{d x}=x y+2 x$. This is her work:$\newline$$$\frac{d y}{d x}=x y+2 x$$$\newline$Step $1$: $\quad \frac{d y}{d x}=x(y+2)$$\newline$Step $2$: $\quad \int(y+2)^{-1} d y=\int x d x$$\newline$Step $3$: $\quad \ln |y+2|=\frac{x^{2}}{2}+C_{1}$$\newline$Step $4$: $\quad e^{\ln |y+2|}=e^{\frac{x^{2}}{2}+C_{1}}$$\newline$Step $5$: $\quad|y+2|=e^{\frac{x^{2}}{2}} e^{C_{1}}$$\newline$Step $6$: $\quad y+2=C e^{\frac{x^{2}}{2}}$$\newline$Step $7$: $\quad y=C e^{\frac{x^{2}}{2}}-2$$\newline$Is Lara's work correct? If not, what is her mistake?$\newline$Choose $1$ answer:$\newline$(A) Lara's work is correct.$\newline$(B) Step $1$ is incorrect. We're not allowed to factor an expression when solving differential equations.$\newline$(C) Step $\mathbf{2}$ is incorrect. The separation of variables wasn't done correctly.$\newline$(D) Step $4$ is incorrect. The right-hand side of the equation should be $e^{\frac{z^{2}}{2}}+C_{1}$.