[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]

Find the general solution of each of the following differential equations.

• [a [math]\dydx = x^3 + 2e^x[/math]
• [math]x\dydx = 6x^3 +5x +1[/math]
• [math]\dydx = (y^2+1)(2x+3)[/math]
• [math]\dydx = xy + x[/math]
• [math]2xy^2 + \dydx - 4x^3y^2 = 0[/math]
• [math]y\dydx = \ln x[/math]
• [math]x\dydx = \ln x[/math]
• [math]\dydx + 16y = 0[/math]
• [math]\deriv2y + 16y = 0[/math]
• [math]\deriv2y = 16y[/math]
• [math]y^{\prime\prime} - 19y^{\prime} - 20y = 0[/math]
• [math](D^2 + 10D + 16)y = 0[/math]
• [math]2\deriv2y - 14\dydx = -20y[/math]
• [math]\deriv2y + a^2y = 2a\dydx[/math]
• [math](D^2 + 4D + 29)y = 0[/math]
• [math](y+5)\dydx = 7x - e^{-x}[/math]
• [math]\dydx = \frac xy[/math]
• [math]\dydx = \frac yx[/math]
• [math]\dydx = -\frac xy[/math]
• [math]\frac1y \deriv2y = 49[/math]
• [math](3x+4)\;dt + (4t+3)\;dx = 0[/math]
• [math]\dydx = \cot y[/math]
• [math]\frac1t \nxder{}yt = e^{3t^2+4}[/math]
• [math]\dydx = 3 \sin^2 x \cos^2 x[/math]
• [math]\dydx = 3 \sin^2 x \cos^2 y[/math]
• [math]\deriv2y = 6x^2 - 4x + 2[/math].