Answer to Question #275717 in Differential Equations for sebby

Question

Evaluate the following functions in differential operator form.

1. D(x²+2x-3) Ans. (2x+2)
2. D²(xe^3x-e^4x) Ans. 9xe^3x+6e^3x-16e^4x

Answer 1

$1.\ D\left(x^{2}+2 x+3\right)$

Since $D=\frac{d}{d x}$

So, we have,

 $\begin{aligned} &\frac{d}{d x}\left(x^{2}+2 x+3\right) \\ &=\frac{d}{d x}\left(x^{2}\right)+\frac{d}{d x}(2 x)+\frac{d}{d x}(3) \\ &=2 x+2 \\ &2.\ D^{2}\left(x e^{3 x}-e^{4 x}\right) \end{aligned}$  

Since $D=\frac{d}{d x}$

So, we have,

 $\begin{aligned} &\frac{d^{2}}{d x^{2}}\left(x e^{3 x}-e^{4 x}\right) \\ &\frac{d}{d x}\left(\frac{d}{d x}\left(x e^{3 x}-e^{4 x}\right)\right) \end{aligned}$

$=\frac{d}{d x}\left(\frac{d}{d x} x e^{3 x}-\frac{d}{d x} e^{4 x}\right)$  

Recall the product rule of differentiation

 $U V=V d u+U d v$

By comparison let $u=x, v=e^{3 x}$

Upon differentiation. We have,

 $\begin{aligned} &\frac{d}{d x}\left(e^{3 x}+3 x e^{3 x}-4 e^{4 x}\right) \\ &=\frac{d}{d x}\left(e^{3 x}\right)+3 \frac{d}{d x}\left(x e^{3 x}\right)-4 \frac{d}{d x}\left(e^{4 x}\right) \\ &=3 e^{3 x}+3\left(e^{3 x}+3 x e^{3 x}\right)-4\left(4 e^{4 x}\right) \\ &=3 e^{3 x}+3 e^{3 x}+9 x e^{3 x}-16 e^{4 x} \\ &=9 x e^{3 x}+6 e^{3 x}-16 e^{4 x} \end{aligned}$