Chapter 1 - First-Order Differential Equations - 1.11 Some Higher-Order Differential Equations - Problems - Page 105: 1

$y(x)=2e^{3x}+C_2e^{2x}+C_1$

Suppose that $a= \dfrac{dy}{dx}$ and $ \dfrac{da}{dx}= \dfrac{d^2y}{dx^2}$ Integrating factor; $I.P.=e^{\int-2 dx}=e^{-2x}$ Now, $\dfrac{d}{dx}(ae^{-2x})=6e^x$ Integrate to obtain: $ae^{-2x}=6e^x+C \implies \dfrac{dy}{dx}=(6e^x+C)e^{2x}$ Therefore, the general solution is: $y(x)=2e^{3x}+C_2e^{2x}+C_1$