(Solved): (2) (a) Solve the following differential equation using separation of variables. (Show your work) \ ...

(2) (a) Solve the following differential equation using separation of variables. (Show your work) \[ \begin{array}{l} y^{\prime}-2 y=0 \\ \frac{y^{\prime}}{y}=L \\ \ln y=2 x+C_{1} \\ y=e^{2 x+C_{1}} \\ y=C_{e}^{2 x} \end{array} \] \[ y^{\prime}-2 y=0 \] (b) Now valve the followitig differential equation. (Show your work) \[ \begin{array}{c} y^{\prime}-2 y=5 \\ y^{\prime}-1 y=5 \\ y^{\prime}=5+2 y \\ \frac{d y}{5+2 y}=d x \\ \frac{1}{2} \ln (1 y+5)=2 x+c_{1} \\ \ln (2 y+f)=2 x+2 c_{1} \\ 2 y=c e^{2 x}-5 \end{array} \] (c) Notion that all we did was change the 0 to a 5 to get froti the first differential equarion to the second. Compare the solution sets. Do you notice anything intesesting (try using the language of Section 2.1).