Learning Guide Learning Guide

6.3 Partial Differential Equations • 265Plotting Partial Differential EquationsThe solutions to many PDEs can be plotted with the PDEplot commandfound in the PDEtools package.> with(PDEtools):You can use the PDEplot command with the following syntax.PDEplot( pde, var, ini, s=range )Here pde is the PDE, var is the dependent variable, ini is a parametriccurve in three-dimensional space with parameter s, and range is the rangeof s.Consider this partial differential equation.> pde := diff(u(x,y), x) + cos(2*x) * diff(u(x,y), y) = -sin(y);pde := ( ∂∂x u(x, y)) + cos(2 x) ( ∂ u(x, y)) = −sin(y)∂yUse the curve given by z = 1 + y 2 as an initial condition, that is,x = 0, y = s, and z = 1 + s 2 .> ini := [0, s, 1+s^2];ini := [0, s, 1 + s 2 ]PDEplot draws the initial-condition curve and the solution surface.> PDEplot( pde, u(x,y), ini, s=-2..2 );u(x,y)71–2–2y22xTo draw the surface, Maple calculates these base characteristic curves.The initial-condition curve is easier to see here than in the previous plot.