Chapter 8 - Mathematical Modeling With Differential Equations - 8.1 Modeling With Differential Equations - Exercises Set 8.1 - Page 567: 17

$c_1 = 2$ $c_2 = -2 $ $y = 2e^{2x} -2xe^{2x} $

$y = c_1e^{2x} + c_2xe^{2x}$ $y' = 2c_1e^{2x} + c_2(2x + 1)e^{2x}$ Plug in the value $y(0) = 2$ $y(0) = c_1e^{2(0)} + c_2(0)e^{2(0)} = 2$ $c_1 = 2$ Plug in the value $y'(0) = 2$ $y'(0) = 2c_1e^{2(0)} + c_2(2(0) + 1)e^{2(0)}= 2$ and $c_1 = 2$ $2(2) + c_2 = 2$ $c_2 = -2$ we get $c_1 = 2$ $c_2 = -2 $ Rewrite the original equation with values of $c_1$ and $c_2$ $y = 2e^{2x} -2xe^{2x} $