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. 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 .