[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]

For each of the following differential equations, a particular solution can be found by inspection. Obtain such a solution [math]y_p[/math], and also find the general solution.

• [math](D^2+3D-10)y=5[/math]
• [math](D^2+1)y = 2x[/math]
• [math]\deriv2y - 4y = 12x - 20[/math]
• [math]\deriv2y + \dydx - 2y = -2x^2+6x-4[/math]
• [math](D^2-2D-3)y=e^x[/math]
• [math]\deriv2y - 2\dydx + y = 8e^{3x}[/math]
• [math]D(D^2-9)y = 2e^{-x}[/math]
• [math](D^2+4)y = 3\sin x[/math]
• [math](D^2+4)y = 3\sin x + 4x + 8[/math]
• [math]\deriv2y + 3\dydx = 5\cos x - 5 \sin x[/math]
• [math](D^2+3)y = 5\cos 3x[/math]
• [math](D^2+2D-2)y = 13 \cos 2x[/math].