Direction Fields

Often it is impossible to find an explicit formula for the solution to a differential equation.  Nevertheless, graphical and numerical approaches may be used to approximate solutions.  In this lesson we will investigate direction fields and the graphical solution.

New Maple Learning
In this lesson we introduce two commands in the package of routines DEtools:
dfieldplot(<eqn>,y(x),x=a..b,y=c..d)  plot the slope field of the differential equation <eqn> in x and y (assuming y is a function of x) for points whose x-coordinates are between a and b and whose y-coordinates are between c and d  NOTE:  We must specify y by y(x) and y' by diff(y(x),x).
dsolve({<eqn>,y(a)=b},y(x))  solves the differential equation <eqn> in x and y (assuming y is a function of x) using the initial condition y(a) = b and y = y(x)

Lesson
Find the direction field associated to the differential y' = x + y and the three solutions corresponding to the initial conditions y(0) = 0, y(0) = 1, and y(0) = -1.

> restart: with(plots): with(DEtools):

> y0:=y(x);y1:=diff(y(x),x);

> plot0:=dfieldplot(y1=x+y0,y0,x=-2..2,y=-2..2):

> display(plot0);

> dsolve({y1=x+y0,y(0)=0},y(x));

> plot1:=plot(rhs(%),x=-2..2,y=-2..2,color=blue):

> display({plot0,plot1});

> dsolve({y1=x+y0,y(0)=1},y(x));

> plot2:=plot(rhs(%),x=-2..2,y=-2..2,color=red):

> display({plot0,plot1,plot2});

> dsolve({y1=x+y0,y(0)=-1},y(x));

> plot3:=plot(rhs(%),x=-2..2,y=-2..2,color=green):

> display({plot0,plot1,plot2,plot3});