[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 particular solution of each of the following differential equations which satisfies the given conditions.

• [math]\dydx = 3y[/math], \quad [math]y=5[/math] when [math]x=0[/math].
• [math]\deriv2y = 12x^2+1[/math], \quad graph passes through the point [math](1,-1)[/math] with a slope of [math]3[/math].
• [math]y\dydx = -x[/math], \quad graph passes through the point [math](-3,-4)[/math].
• [math]\nxder2st = -g \mbox{constant}[/math], \quad when [math]t=0[/math], [math]\nxder{}st = v_0[/math] and [math]s=s_0[/math].
• [math](D^2-2D-3)y = 0[/math], \quad [math]y=7[/math] and [math]\dydx = 1[/math] when [math]x=0[/math].
• [math](D^2-4D+13)y = 0[/math], \quad graph passes through [math](0,5)[/math] with a slope of [math]2[/math].
• [math](x+2)\dydx = 1[/math], \quad [math]y=\ln 9[/math] when [math]x=1[/math].
• [math](D^2-12D+36)y=0[/math], \quad [math]y=3[/math] and [math]\dydx=7[/math] when [math]x=0[/math].