AS/A2 Pure 5 Pure 6

P4 Topic 5: First Order Differential Equations
Families of solutions 4
When solving a differential equation dy/dx = f(x,y), the solution will have an arbitary constant, c. There is therefore a family of solutions that depend on the value of c. In the following displays families of solution curves are shown for some simple examples of dy/dx = f(x,y). You can display any particular solution by clicking at any point on the graph. The x-coordinate of the point you click determines c. [Ignore the first input box.]

Summary Clearly the differential equations given above have these solutions: dy/dx y y ex+c 2xy ex2+c 3x2y ex3+c 2xy2 1/(c-x2) y2 1/(c-x) 1/x ln x + c y/x cx y2/x2 x/(1-cx) Note that the x value you click on is not the value of c itself. It determines where the solution y is zero. All the above examples are where f(x) consists of a product of a function of x with a function of y. Such cases are called separable differential equations. Based on free Java applets from JavaMath