Solve y=c_{1}e^x+c_{2}e^-xe^i?

$$y = c _ { 1 } e ^ { x } + c _ { 2 } e ^ { - x } e ^ { i } ?$$

Solve for c_1

$$c_{1}=\frac{y+c_{2}e^{-x+\pi i+i}}{e^{x}}$$

Steps for Solving Linear Equation

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

$$c_{1}e^{x}+c_{2}e^{-x}e^{i}=y$$

Subtract c_{2}e^{-x}e^{i} from both sides.

$$c_{1}e^{x}=y-c_{2}e^{-x}e^{i}$$

Reorder the terms.

$$c_{1}e^{x}=y-e^{i}c_{2}e^{-x}$$

The equation is in standard form.

$$e^{x}c_{1}=y-c_{2}e^{-x+i}$$

Divide both sides by e^{x}.

$$\frac{e^{x}c_{1}}{e^{x}}=\frac{y-c_{2}e^{-x+i}}{e^{x}}$$

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

$$c_{1}=\frac{y-c_{2}e^{-x+i}}{e^{x}}$$

Solve for c_2

$$c_{2}=ye^{x-i}-c_{1}e^{2x-i}$$

Steps for Solving Linear Equation

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

$$c_{1}e^{x}+c_{2}e^{-x}e^{i}=y$$

Subtract c_{1}e^{x} from both sides.

$$c_{2}e^{-x}e^{i}=y-c_{1}e^{x}$$

Reorder the terms.

$$e^{i}c_{2}e^{-x}=y-c_{1}e^{x}$$

The equation is in standard form.

$$e^{i-x}c_{2}=y-c_{1}e^{x}$$

Divide both sides by e^{-x+i}.

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

Dividing by e^{-x+i} undoes the multiplication by e^{-x+i}.

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