Chapter 8.3, Problem 24ES

(a) Find a slope field whose integral curve through (0, 0) satisfies $x{e}^y+y{e}^x=0$ by differentiating this equation implicitly.

(b) Prove that if $y\left(x\right)$ is any integral curve of the slope field in part (a), then $x{e}^{y\left(x\right)}+y\left(x\right)e^x$ will be a constant function.

(c) Find an equation that implicitly defines the integral curve through $\left(1,1\right)$ of the slope field in part (a).