Solve y^prime=(cx^2+2x+2)e^-3x

$$y ^ { \prime } = ( c x ^ { 2 } + 2 x + 2 ) e ^ { - 3 x }$$

Solve for c

$$c=-\frac{2\left(x+1\right)}{x^{2}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply cx^{2}+2x+2 by e^{-3x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(y)=cx^{2}e^{-3x}+2xe^{-3x}+2e^{-3x}$$

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

$$cx^{2}e^{-3x}+2xe^{-3x}+2e^{-3x}=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Subtract 2xe^{-3x} from both sides.

$$cx^{2}e^{-3x}+2e^{-3x}=\frac{\mathrm{d}}{\mathrm{d}x}(y)-2xe^{-3x}$$

Subtract 2e^{-3x} from both sides.

$$cx^{2}e^{-3x}=\frac{\mathrm{d}}{\mathrm{d}x}(y)-2xe^{-3x}-2e^{-3x}$$

The equation is in standard form.

$$\frac{x^{2}}{e^{3x}}c=\frac{-2x-2}{e^{3x}}$$

Divide both sides by x^{2}e^{-3x}.

$$\frac{\frac{x^{2}}{e^{3x}}ce^{3x}}{x^{2}}=\frac{\left(-\frac{2\left(x+1\right)}{e^{3x}}\right)e^{3x}}{x^{2}}$$

Dividing by x^{2}e^{-3x} undoes the multiplication by x^{2}e^{-3x}.

$$c=\frac{\left(-\frac{2\left(x+1\right)}{e^{3x}}\right)e^{3x}}{x^{2}}$$

Divide -\frac{2\left(1+x\right)}{e^{3x}} by x^{2}e^{-3x}.

$$c=-\frac{2\left(x+1\right)}{x^{2}}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{1-2c}+1}{c}\text{; }x=\frac{\sqrt{1-2c}-1}{c}\text{, }&c\neq 0\text{ and }c\leq \frac{1}{2}\\x=-1\text{, }&c=0\end{matrix}\right.$$