Solve y=C_{1}x+C_{2}e^x

$$y = C _ { 1 } x + C _ { 2 } e ^ { x }$$

Solve for C_1 (complex solution)

$$\left\{\begin{matrix}C_{1}=-\frac{C_{2}e^{x}-y}{x}\text{, }&x\neq 0\\C_{1}\in \mathrm{C}\text{, }&y=C_{2}\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$C_{1}x+C_{2}e^{x}=y$$

Subtract C_{2}e^{x} from both sides.

$$C_{1}x=y-C_{2}e^{x}$$

The equation is in standard form.

$$xC_{1}=y-C_{2}e^{x}$$

Divide both sides by x.

$$\frac{xC_{1}}{x}=\frac{y-C_{2}e^{x}}{x}$$

Dividing by x undoes the multiplication by x.

$$C_{1}=\frac{y-C_{2}e^{x}}{x}$$

Solve for C_1

$$\left\{\begin{matrix}C_{1}=-\frac{C_{2}e^{x}-y}{x}\text{, }&x\neq 0\\C_{1}\in \mathrm{R}\text{, }&y=C_{2}\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$C_{1}x+C_{2}e^{x}=y$$

Subtract C_{2}e^{x} from both sides.

$$C_{1}x=y-C_{2}e^{x}$$

The equation is in standard form.

$$xC_{1}=y-C_{2}e^{x}$$

Divide both sides by x.

$$\frac{xC_{1}}{x}=\frac{y-C_{2}e^{x}}{x}$$

Dividing by x undoes the multiplication by x.

$$C_{1}=\frac{y-C_{2}e^{x}}{x}$$

Solve for C_2

$$C_{2}=-\frac{C_{1}x-y}{e^{x}}$$

Steps for Solving Linear Equation

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

$$C_{1}x+C_{2}e^{x}=y$$

Subtract C_{1}x from both sides.

$$C_{2}e^{x}=y-C_{1}x$$

The equation is in standard form.

$$e^{x}C_{2}=y-C_{1}x$$

Divide both sides by e^{x}.

$$\frac{e^{x}C_{2}}{e^{x}}=\frac{y-C_{1}x}{e^{x}}$$

Dividing by e^{x} undoes the multiplication by e^{x}.

$$C_{2}=\frac{y-C_{1}x}{e^{x}}$$