You can use PGFPlots' quiver
plot style for drawing the vector fields.
I'm not entirely sure what you mean by "all possible solution curves", since that would just cover the whole plot area. I just drew one possible solution for each equation, all others would just be vertically shifted versions:

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.8}
\usepackage{amsmath}
\pgfplotsset{ % Define a common style, so we don't repeat ourselves
MaoYiyi/.style={
width=0.6\textwidth, % Overall width of the plot
axis equal image, % Unit vectors for both axes have the same length
view={0}{90}, % We need to use "3D" plots, but we set the view so we look at them from straight up
xmin=0, xmax=1.1, % Axis limits
ymin=0, ymax=1.1,
domain=0:1, y domain=0:1, % Domain over which to evaluate the functions
xtick={0,0.5,1}, ytick={0,0.5,1}, % Tick marks
samples=11, % How many arrows?
cycle list={ % Plot styles
gray,
quiver={
u={1}, v={f(x)}, % End points of the arrows
scale arrows=0.075,
every arrow/.append style={
-latex % Arrow tip
},
}\\
red, samples=31, smooth, thick, no markers, domain=0:1.1\\ % The plot style for the function
}
}
}
\begin{document}
\begin{tikzpicture}[
declare function={f(\x) = 2*\x;} % Define which function we're using
]
\begin{axis}[
MaoYiyi, title={$\dfrac{\mathrm{d}y}{\mathrm{d}x}=2x$}
]
\addplot3 (x,y,0);
\addplot {x^2+0.15}; % You need to find the antiderivative yourself, unfortunately. Good exercise!
\end{axis}
\end{tikzpicture}
%
\begin{tikzpicture}[
declare function={f(\x) = \x*sqrt(\x);}
]
\begin{axis}[
MaoYiyi,
title={$\dfrac{\mathrm{d}y}{\mathrm{d}x}=x\sqrt{x}$},
ytick=\empty
]
\addplot3 (x,y,0);
\addplot +[domain=0.001:1.1] {x^(2.5)/2.5+0.15};
\end{axis}
\end{tikzpicture}
\end{document}
pst-ode
example is here: tex.stackexchange.com/a/139140 – AlexG Oct 16 '13 at 14:37