Answer
$$y = - 5{e^{ - x}} - 5x + 22$$
Work Step by Step
$$\eqalign{
& \frac{{dy}}{{dx}} = 5\left( {{e^{ - x}} - 1} \right);\,\,\,\,\,\,\,\,\,y\left( 0 \right) = 17 \cr
& {\text{Separating variables leads to}} \cr
& dy = 5\left( {{e^{ - x}} - 1} \right)dx \cr
& {\text{To solve this equation}}{\text{, determine the antiderivative of each side}} \cr
& \int {dy} = \int {5\left( {{e^{ - x}} - 1} \right)dx} \cr
& \int {dy} = \int {\left( {5{e^{ - x}} - 5} \right)dx} \cr
& {\text{integrating}} \cr
& y = - 5{e^{ - x}} - 5x + C\,\,\,\,\,\left( {\bf{1}} \right) \cr
& \cr
& {\text{we can find the constant }}C{\text{ using the initial value problem}}\,\,y\left( 0 \right) = 17 \cr
& y\left( 0 \right) = 17{\text{ implies that }}y = 17{\text{ when }}x = 0 \cr
& {\text{substituting these values into }}\left( {\bf{1}} \right) \cr
& 17 = - 5{e^{ - 0}} - 5\left( 0 \right) + C \cr
& C = 17 + 5 \cr
& C = 22 \cr
& {\text{substitute }}C = 3{\text{ into }}\left( {\bf{1}} \right){\text{ to find the particular solution}} \cr
& y = - 5{e^{ - x}} - 5x + 22 \cr} $$
