Solve y=e^-xy_{2}=2ex^-2quady_{b}=3e^2

$$y = e ^ { - x } y _ { 2 } = 2 e x ^ { - 2 } \quad y _ { b } = 3 e ^ { 2 }$$

Evaluate (complex solution)

$$y=\frac{y_{2}}{e^{x}}\text{ and }\frac{y_{2}}{e^{x}}=\frac{2ey_{b}}{x^{2}}\text{ and }\frac{2ey_{b}}{x^{2}}=3e^{2}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{\frac{6y_{b}}{e}}}{3}\text{, }&y=3e^{2}\text{ and }y_{b}=\frac{3e\left(\ln(y_{2})-\ln(3)-2\right)^{2}}{2}\text{ and }y_{2}>3e^{2}\\x=-\frac{\sqrt{\frac{6y_{b}}{e}}}{3}\text{, }&y=3e^{2}\text{ and }y_{b}=\frac{3e\left(-\ln(y_{2})+\ln(3)+2\right)^{2}}{2}\text{ and }y_{2}>0\text{ and }y_{2}<3e^{2}\end{matrix}\right.$$

Solve for y_b

$$y_{b}=\frac{3e\left(\ln(y_{2})-\ln(3)-2\right)^{2}}{2}$$

$$y=3e^{2}\text{ and }x=\ln(y_{2})-\ln(3)-2\text{ and }y_{2}>0\text{ and }y_{2}\neq 3e^{2}$$