Solve [37]f(x)=xe^-x

$$[ 37 ] f ( x ) = x e ^ { - x }$$

Solve for f (complex solution)

$$\left\{\begin{matrix}\\f=\frac{1}{37e^{x}}\text{, }&\text{unconditionally}\\f\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$37xf=\frac{x}{e^{x}}$$

Divide both sides by 37x.

$$\frac{37xf}{37x}=\frac{x}{e^{x}\times 37x}$$

Dividing by 37x undoes the multiplication by 37x.

$$f=\frac{x}{e^{x}\times 37x}$$

Divide \frac{x}{e^{x}} by 37x.

$$f=\frac{1}{37e^{x}}$$

Solve for f

$$\left\{\begin{matrix}\\f=\frac{1}{37e^{x}}\text{, }&\text{unconditionally}\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$37xf=\frac{x}{e^{x}}$$

Divide both sides by 37x.

$$\frac{37xf}{37x}=\frac{x}{e^{x}\times 37x}$$

Dividing by 37x undoes the multiplication by 37x.

$$f=\frac{x}{e^{x}\times 37x}$$

Divide \frac{x}{e^{x}} by 37x.

$$f=\frac{1}{37e^{x}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=-\left(\ln(f)+\ln(37)\right)+2\pi n_{1}i\text{, }n_{1}\in \mathrm{Z}\text{, }&f\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=-\left(\ln(f)+\ln(37)\right)\text{, }&f>0\end{matrix}\right.$$