Solve y=Ax+x^2e^-lambda

$$y = A x + x ^ { 2 } e ^ { - \lambda }$$

Solve for A (complex solution)

$$\left\{\begin{matrix}A=-\frac{x}{e^{\lambda }}+\frac{y}{x}\text{, }&x\neq 0\\A\in \mathrm{C}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Ax+x^{2}e^{-\lambda }=y$$

Subtract x^{2}e^{-\lambda } from both sides.

$$Ax=y-x^{2}e^{-\lambda }$$

Reorder the terms.

$$Ax=-x^{2}e^{-\lambda }+y$$

The equation is in standard form.

$$xA=-\frac{x^{2}}{e^{\lambda }}+y$$

Divide both sides by x.

$$\frac{xA}{x}=\frac{ye^{\lambda }-x^{2}}{e^{\lambda }x}$$

Dividing by x undoes the multiplication by x.

$$A=\frac{ye^{\lambda }-x^{2}}{e^{\lambda }x}$$

Divide \frac{ye^{\lambda }-x^{2}}{e^{\lambda }} by x.

$$A=\frac{ye^{\lambda }-x^{2}}{xe^{\lambda }}$$

Solve for A

$$\left\{\begin{matrix}A=-\frac{x}{e^{\lambda }}+\frac{y}{x}\text{, }&x\neq 0\\A\in \mathrm{R}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Ax+x^{2}e^{-\lambda }=y$$

Subtract x^{2}e^{-\lambda } from both sides.

$$Ax=y-x^{2}e^{-\lambda }$$

Reorder the terms.

$$Ax=-x^{2}e^{-\lambda }+y$$

The equation is in standard form.

$$xA=-\frac{x^{2}}{e^{\lambda }}+y$$

Divide both sides by x.

$$\frac{xA}{x}=\frac{ye^{\lambda }-x^{2}}{e^{\lambda }x}$$

Dividing by x undoes the multiplication by x.

$$A=\frac{ye^{\lambda }-x^{2}}{e^{\lambda }x}$$

Divide \frac{ye^{\lambda }-x^{2}}{e^{\lambda }} by x.

$$A=\frac{ye^{\lambda }-x^{2}}{xe^{\lambda }}$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{e^{\lambda }\left(4y+A^{2}e^{\lambda }\right)}-Ae^{\lambda }}{2}$$

$$x=\frac{\sqrt{e^{\lambda }\left(4y+A^{2}e^{\lambda }\right)}-Ae^{\lambda }}{2}$$

Solve for x

$$x=\frac{-e^{\frac{\lambda }{2}}\sqrt{4y+A^{2}e^{\lambda }}-Ae^{\lambda }}{2}$$

$$x=\frac{e^{\frac{\lambda }{2}}\sqrt{4y+A^{2}e^{\lambda }}-Ae^{\lambda }}{2}\text{, }y\geq -\frac{A^{2}e^{\lambda }}{4}$$