Directional field and the initial value problem
- Przemysław Kajetanowicz
The applet above illustrates the concept of the directional field of an ordinary differential equation and of the initial value problem. The initial value problem is: find the solution of the equation that satisfies for given . The last condition is equvalent to saying that the graph of the desired solution passes through . Use the mouse to move the point within the available area of the coordinate system. Geogebra recalculates the solution and displays its graph accordingly. Note: on some computers things may behave tardily. Also, this web version is slower than the original file opened in Geogebra. When the page loads, the default example shows the solution of with the initial condition . A general solution of the equation is the family of circles . This can be determined by appropriate calculation, but also guessed by inspecting the shape of the directional field. When the initial condition is additionally imposed, we get , hence . Solving the equation in and allowing for the fact that yields the solution . Note that the domain of the solution is the interval . Feel free to experiment with your own expressions .