Chapter 7 - Section 7.2 - Exponential Change and Separable Differential Equations - Exercises - Page 409: 1

a) $e^{-x}$ b) $e^{-x}$ c) $e^{-x}$

a) since, $y'=-e^{-x}$ Thus, $2y'+3y=2 (-e^{-x}) +3 e^{-x}=e^{-x}$ b) Since, $y'=-e^{-x}-\dfrac{3}{2}e^{-\frac{3}{2}x}$ Thus, $2y'+3y=2 (-e^{-x}) -2[\dfrac{3}{2}e^{-\frac{3}{2}x}]+3 e^{-x}+3e^{-\frac{3}{2}x}=e^{-x}$ c) Since, $y'=-e^{-x}-\dfrac{3}{2} Ce^{-\frac{3}{2}x}$ Thus, $2y'+3y=2 (-e^{-x}) -2[\dfrac{3}{2}Ce^{-\frac{3}{2}x}]+3 e^{-x}+3Ce^{-\frac{3}{2}x}=e^{-x}$