Solve y=(c_{1}+c_{2}x)e^-frac{1{2}x}

$$y = ( c _ { 1 } + c _ { 2 } x ) e ^ { - \frac { 1 } { 2 } x }$$

Solve for c_1

$$c_{1}=ye^{\frac{x}{2}}-c_{2}x$$

Steps for Solving Linear Equation

Use the distributive property to multiply c_{1}+c_{2}x by e^{-\frac{1}{2}x}.

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

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

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

Subtract c_{2}xe^{-\frac{1}{2}x} from both sides.

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

Reorder the terms.

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

The equation is in standard form.

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

Divide both sides by e^{-\frac{1}{2}x}.

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

Dividing by e^{-\frac{1}{2}x} undoes the multiplication by e^{-\frac{1}{2}x}.

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

Divide y-e^{-\frac{x}{2}}c_{2}x by e^{-\frac{1}{2}x}.

$$c_{1}=ye^{\frac{x}{2}}-c_{2}x$$

Solve for c_2

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

Steps for Solving Linear Equation

Use the distributive property to multiply c_{1}+c_{2}x by e^{-\frac{1}{2}x}.

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

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

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

Subtract c_{1}e^{-\frac{1}{2}x} from both sides.

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

The equation is in standard form.

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

Divide both sides by xe^{-\frac{1}{2}x}.

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

Dividing by xe^{-\frac{1}{2}x} undoes the multiplication by xe^{-\frac{1}{2}x}.

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

Divide y-e^{-\frac{x}{2}}c_{1} by xe^{-\frac{1}{2}x}.

$$c_{2}=\frac{ye^{\frac{x}{2}}-c_{1}}{x}$$