First-Order Ordinary Linear Differential Equations in one variable (dy/dx = f(x)) are easy to solve using the antiderivative to find the function y = F(x) + C, and then applying the initial condition (x0, y0) to find the value of the constant of integration C. As we move to more complicated types of differential equations, the solution methods also become more complicated. However, if we are considering equations of the form dy/dx = f(x, y), we can use a graphical method to estimate solutions.
Note that one interpretation of the equation dy/dx = f(x, y) is "the slope of the solution y is equal to the value of f(x, y) at any point (x, y)". Example: If the differential equation is dy/dx = x + y, we know that the solution's slope is 0 at the origin, 2 at (1, 1), 6 at (4, 2), -3 at (4, -7), and so on. By sketching short segments at many points (x, y) having slope f(x, y), a graphical solution arises. The graph of the solution passing through the initial condition (x0, y0) will follow these segment slopes away from (x0, y0).
If you look only at the slope field (not including the graph of the solution), you can visualize the shape of the solution passing through any given (x0, y0) in the field. Although it is true that a solution graph passing through any plotted segment will have the same slope as the segment at that point, be careful not to assume that the desired solution will follow only the plotted segments - it's likely to weave in between them. A "denser" slope field (i.e., smaller spacing) will offer insight into local behavior of the solution graph.

Type in a function in x and/or y in the dy/dx input box. The slope field for that function in the window [(-4, -4) , (4, 4)] will be plotted. You can adjust the spacing (density) and the lengths of the segments with the appropriate sliders. Drag the large red point to the coordinate representing the desired intitial condition (x0, y0) to see a plot of the particular solution. Note that the graph has the same slope as the segments at any point where a segment is plotted, but also note that the solution is likely to move in a somewhat different pattern than might be suggested looking only at the segments.
Note also that the method used to graph the solution (Euler's Method) can fail if it encounters a point of vertical slope. If the graph suddenly takes on a random, spiky character, it's like that the computation has hit such a snag.