Solve frac{dC_{2}(t)}{dt}+k_{e}C_{2}(t)=k_{1}C_{0}e^-k_{1t}

$$\frac { d C _ { 2 } ( t ) } { d t } + k _ { e } C _ { 2 } ( t ) = k _ { 1 } C _ { 0 } e ^ { - k _ { 1 } t }$$

Solve for k

$$k=C_{2}\left(k_{e}t+1\right)e^{k_{1}t}$$

Steps for Solving Linear Equation

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

$$kcombination(1,0)e^{\left(-k_{1}\right)t}=\frac{\mathrm{d}(C_{2}t)}{\mathrm{d}t}+k_{e}C_{2}t$$

Reorder the terms.

$$ke^{-k_{1}t}combination(1,0)=\frac{\mathrm{d}(C_{2}t)}{\mathrm{d}t}+C_{2}k_{e}t$$

The equation is in standard form.

$$\frac{1}{e^{k_{1}t}}k=C_{2}k_{e}t+C_{2}$$

Divide both sides by e^{-k_{1}t}.

$$\frac{\frac{1}{e^{k_{1}t}}ke^{k_{1}t}}{1}=\frac{\left(C_{2}k_{e}t+C_{2}\right)e^{k_{1}t}}{1}$$

Dividing by e^{-k_{1}t} undoes the multiplication by e^{-k_{1}t}.

$$k=\frac{\left(C_{2}k_{e}t+C_{2}\right)e^{k_{1}t}}{1}$$

Divide C_{2}+C_{2}k_{e}t by e^{-k_{1}t}.

$$k=C_{2}\left(k_{e}t+1\right)e^{k_{1}t}$$

Solve for C_2

$$\left\{\begin{matrix}C_{2}=\frac{k}{\left(k_{e}t+1\right)e^{k_{1}t}}\text{, }&k_{e}=0\text{ or }t\neq -\frac{1}{k_{e}}\\C_{2}\in \mathrm{R}\text{, }&k=0\text{ and }t=-\frac{1}{k_{e}}\text{ and }k_{e}\neq 0\end{matrix}\right.$$