Solve y=c_{1}+c_{2}x+c_{3}x^2+c_{4}e^2x

$$y = c _ { 1 } + c _ { 2 } x + c _ { 3 } x ^ { 2 } + c _ { 4 } e ^ { 2 x }$$

Solve for c_2 (complex solution)

$$\left\{\begin{matrix}c_{2}=-\frac{c_{3}x^{2}-y+c_{4}e^{2x}+c_{1}}{x}\text{, }&x\neq 0\\c_{2}\in \mathrm{C}\text{, }&y=c_{1}+c_{4}\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$c_{1}+c_{2}x+c_{3}x^{2}+c_{4}e^{2x}=y$$

Subtract c_{1} from both sides.

$$c_{2}x+c_{3}x^{2}+c_{4}e^{2x}=y-c_{1}$$

Subtract c_{3}x^{2} from both sides.

$$c_{2}x+c_{4}e^{2x}=y-c_{1}-c_{3}x^{2}$$

Subtract c_{4}e^{2x} from both sides.

$$c_{2}x=y-c_{1}-c_{3}x^{2}-c_{4}e^{2x}$$

Reorder the terms.

$$c_{2}x=-c_{3}x^{2}+y-c_{4}e^{2x}-c_{1}$$

The equation is in standard form.

$$xc_{2}=-c_{3}x^{2}+y-c_{4}e^{2x}-c_{1}$$

Divide both sides by x.

$$\frac{xc_{2}}{x}=\frac{-c_{3}x^{2}+y-c_{4}e^{2x}-c_{1}}{x}$$

Dividing by x undoes the multiplication by x.

$$c_{2}=\frac{-c_{3}x^{2}+y-c_{4}e^{2x}-c_{1}}{x}$$

Solve for c_2

$$\left\{\begin{matrix}c_{2}=-\frac{c_{3}x^{2}-y+c_{4}e^{2x}+c_{1}}{x}\text{, }&x\neq 0\\c_{2}\in \mathrm{R}\text{, }&y=c_{1}+c_{4}\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$c_{1}+c_{2}x+c_{3}x^{2}+c_{4}e^{2x}=y$$

Subtract c_{1} from both sides.

$$c_{2}x+c_{3}x^{2}+c_{4}e^{2x}=y-c_{1}$$

Subtract c_{3}x^{2} from both sides.

$$c_{2}x+c_{4}e^{2x}=y-c_{1}-c_{3}x^{2}$$

Subtract c_{4}e^{2x} from both sides.

$$c_{2}x=y-c_{1}-c_{3}x^{2}-c_{4}e^{2x}$$

Reorder the terms.

$$c_{2}x=-c_{3}x^{2}+y-c_{4}e^{2x}-c_{1}$$

The equation is in standard form.

$$xc_{2}=-c_{3}x^{2}+y-c_{4}e^{2x}-c_{1}$$

Divide both sides by x.

$$\frac{xc_{2}}{x}=\frac{-c_{3}x^{2}+y-c_{4}e^{2x}-c_{1}}{x}$$

Dividing by x undoes the multiplication by x.

$$c_{2}=\frac{-c_{3}x^{2}+y-c_{4}e^{2x}-c_{1}}{x}$$

Solve for c_1

$$c_{1}=-c_{3}x^{2}-c_{2}x+y-c_{4}e^{2x}$$