Chapter 8 - Mathematical Modeling With Differential Equations - 8.1 Modeling With Differential Equations - Exercises Set 8.1 - Page 567: 18

$c_1 = 1$ $c_2 = -3 $ $y = e^{4x} -3 xe^{4x}$

$y = c_1e^{4x} +c_2xe^{4x} $ $y' = 4c_1e^{4x} +c_2(4x+1)e^{4x} $ Plug in the value $y(0) = 1$ $y(0) = c_1e^{4(0)} +c_2(0)e^{4(0)} = 1$ $c_1 = 1 ............... (1)$ Plug in the value $y'(0) = 1$ $y'(0) = 4c_1e^{4(0)} +c_2(4(0)+1)e^{4(0)} = 1$ and $c_1 = 1$ $4(1) + c_2 = 1..............(2)$ we get $c_1 = 1$ $c_2 = -3 $ Rewrite the original equation with values of $c_1$ and $c_2$ $y = e^{4x} -3 xe^{4x}$