[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

Using Theorem \ref{thm 11.4.4}, which gives the general real-valued solution of the [math]n[/math]th-order differential equation [math]p(D)y=0[/math], solve each of the following.

• [math](D-2)(D+1)^2y=0[/math]
• [math]\deriv3y - 7\dydx + 6y = 0[/math]
• [math](D-3)^2(D+1)(D-5)y=0[/math]
• [math]D(D^2+3D-4)y=0[/math]
• [math](D+2)^3(D-1)y=0[/math]
• [math](D+3)^2(D^2+3)y=0[/math]
• [math]\deriv3y + \deriv2y - 2\dydx = 0[/math]
• [math](D^2+2D+2)^2y=0[/math]
• [math](D+1)(D^2+2D+2)^2y=0[/math]
• [math]D^2(D^2+2D+2)^2y=0[/math]
• [math]\deriv4y - 81y = 0[/math]
• [math]\deriv3y+\deriv2y+\dydx+y=0[/math].