Question

perform this using the linear differential equation of higher order in operator form: (D2 + 3D + 2) (e^-2x + 3x²)

Answer 1

$\begin{aligned} &\left(D^{2}+3 D+2\right)\left(e^{x}-2 x+3 x^{2}\right)\\ &=D^{2}\left(e^{x}-2 x+3 x^{2}\right)+3 D\left(e^{x}-2 x+3 n^{2}\right) +2\left(e^{x}-2 x+3 x^{2}\right)\\ &= \frac{d^{2}}{d x^{2}}\left(e^{x}-2 x+3 x^{2}\right)+3 \frac{d}{d x}\left(e^{x}-2 x+3 x^{2}\right) +2\left(e^{x}-2 x+3 x^{2}\right)\\ &=\frac{d}{d x}\left[\frac{d}{d x}\left(e^{x}-2 x+3 x^{2}\right)\right]+3\left(e^{x}-2+6 x\right) +2\left(e^{x}-2 x+3 x^{2}\right)\\ &=\frac{d}{d x}\left[e^{x}-2+6 x\right]+3 e^{x}-6+18 x+2 e^{x}-4 x+6 x^{2}\\ &=e^{x}+6+3 e^{x}-6+18 x+2 e^{x}-4 x+6 x^{2}\\ &=6 e^{x}+14 x+6 x^{2} \end{aligned}$