[b]Background information:[/b][br][br]First order differential equations with initial values of the form [br][br][math]\frac{dy}{dx}=f\left(x,y\right);y\left(x_0\right)=y_0[/math][br][br]may or may not have specific algebraic solutions depending on the form of [math]f(x,y)[/math]. Special forms of [math]f(x,y)[/math] that admit algebraic solutions include, linear, exact, and separable. This list is not exhaustive.[br][br]Regardless of the form of [math]f(x,y)[/math] however, numerical methods of solving these types of first order initial value problems are always available. The most well known method of advancing through the independent variable systematically is Euler's Method. Euler's Method for advancing from [math](x_0,y_0)[/math] is plotted in orange in the applet below. Euler's Method, however, frequently makes systematic errors by "overshooting" important information contained in [math]f(x,y)[/math]. For instance, below, the differential equation [math]\frac{dy}{dx}=\frac{-x}{y}[/math]has algebraic solutions [math]x^2+y^2=c[/math], circles, but Euler's Method will generate outward spirals because it always "overshoots" [math]f(x,y)[/math]. [br][br]A famous improvement on Euler's Method is known as the Runge Kutta (RK) family of methods. There are infinitely many methods in the RK Family, and in fact Euler's Method is the "first" member of family. The most widely used member method of the RK family is version 4, and it is plotted as the blue arrow below. [br][br]This interactive activity illustrates one step of the RK4 method for estimating the forward value, [math](x_1,y_1)[/math], of a first order initial value problem like above. In order to advance further, RK4 is repeated with using the forward value [math](x_1,y_1)[/math] as new initialization data to produce [math](x_2,y_2)[/math]. The process then repeats for as long as the user demands.[br][br][b]To use this illustrative applet:[/b][br][br]Adjust (or don't) the differential equation in the input box in the top right.[br][br]Adjust (or don't) the step size, and the initial condition [math](x_0,y_0)[/math]. Initial conditions should be entered as points ([i]i.e.[/i] in parentheses, separated by a comma). The initial condition can also be adjusted by dragging [math](x_0,y_0)[/math] in the graphics pane on the left. [br][br][b]What you're looking at:[/b][br][br]The four component equations of the Runge Kutta method, [math]k_1,k_2,k_3[/math], and [math]k_4[/math] are illustrated in orange, purple, pink and red, respectively. These four components can be thought of as 4 "soundings" of the differential equation's slope field starting from [math](x_0,y_0)[/math]. The 4 soundings are systematically made according to the RK4 equations. [br][br]The RK Trajectory (blue) illustrates the RK4 estimate of how to advance from initial condition [math](x_0,y_0)[/math] to [math](x_1,y_1)[/math] using a weighted average of the four soundings. [br][br]For those who wish to know a little more: The horizontal displacement of RK Trajectory, like Euler's Method, is the step size. The vertical displacement is the weighted average of the vertical components of the 4 soundings, [math](k_1+2k_2+2k_3+k_4)[/math].

Check back later for an additional applet illustrating the process by which RK4 is iterated to advance to additional forward values [math]\left(x_n,y_n\right)[/math].