Setup.with(plots): with(DEtools):A sample second order equationde0:=diff(y(x),x,x)-y(x)=x^3;ic0:=y(0)=1,D(y)(0)=2;dsolve(de0);dsolve({de0,ic0});Solving a system of equationsHigher order equations such as de1 may also be converted to first order systems in which y and some of its derivatives are taken as separate variables.ds0:=diff(y(x),x)=y1(x),diff(y1(x),x)=y(x)+x^3;dsolve({ds0});ics0 := y(0)=1, y1(0)=2;dsolve({ds0,ics0});Basic tools: infolevel and odeadvisorodeadvisor(de0);odeadvisor({ds0});#this will fail because only single equations are valid input.infolevel[dsolve]:=2;dsolve(de0);infolevel[dsolve]:=1;Interactive toolsdsolve[interactive]({de0,ic0});PlotsDEplot(de0,y(x),x=-2..2,[[ic0]], title="First equation");ic0a:=y(0)=1,D(y)(0)=0;DEplot(de0, y(x), x = -2 .. 2, [[ic0],[ic0a]], title="A pair of solutions");A first order equationde1:= diff(y(x),x) = - y(x) + 1/(1 + exp(x)); Finding the general solutions:ans1:=dsolve(de1,y(x));#ans1 denotes an equations1:=eval(y(x),ans1); #the solution can be recovered Finding the solution with an initial condition:ic1 := y(0)=2;ans1a:=dsolve({de1,ic1},y(x));s1a:=eval(y(x),ans1a); Plotting commandsPlotting only the direction field:DEplot(de1, y(x), x=-1..5,y=-6..4);Plotting the direction field with two numerical solutions:initval:={[y(0)=-2],[y(0)=1]};DEplot(de1, y(x), x=-1..5, initval,y=-6..4);Use of other commandsThe next command plots the explicit solution found above:plot(s1a,x=-1..5);Now we find a numerical solution with the same initial condition y(0(=2:s1n:=dsolve({de1,y(0)=2},y(x),numeric);The plot of the numerical should look the same as the plot found above:odeplot(s1n,[x,y(x)],-1..5);Finally we plot the direction field:dfieldplot(de1,y(x), x=-1..5,y= -6..4);