Homogenit
Kar. elev.: $r^2 + r - 2 = 0 \Rightarrow r = 1$ oder $r = -2$
$y_n = C_1 e^{x} + C_2 e^{-2x}$
Partikular
Anstatt $y = ax^2 + bx + c \Rightarrow y^1 = 2ax + b \Rightarrow y^0 = 2a$
$\Rightarrow 2x^2 + 3x + 1 = y^0 + y^1 - 2y = 2a + 2ax + b - 2(ax^2 + bx + c) =$
$= -2ax^2 + (2a - 2b)x + 2a + b - 2c$
$\Rightarrow

\begin{cases} -2a = 2 \\ 2a - 2b = 3 \\ 2a + b - 2c = 1 \end{cases}

\Rightarrow

\begin{cases} a = -1 \\ b = \frac{2a - 3}{2} = -\frac{5}{2} \\ c = \frac{2a + b - 1}{2} = -\frac{3 - \frac{5}{2}}{2} = -\frac{11}{4} \end{cases}

$ $\Rightarrow y = y_n + y_p = C_1 e^{x} + C_2 e^{-2x} - x^2 - \frac{5}{2}x - \frac{11}{4}$