nullnullSlope Fields 斜率场 Université LavalnullThe differential equationtells us that at the point (x, y) on the solution curve y=y(x), the slope of the curve is f(x, y).null If we draw short line segments with slope f(x,y) at point (x,y), then we obtain a slope field (or direction field) for the differential equation. These line segments indicate the direction in which a solution curve is heading, so the slope filed helps us visualize the general shape of the solution curve.nullFor example The differential equation tells us that the slope of the solution curve y=y(x) at point (x, y) is 2xy.Point Slope (1,1) 2 (-1,2) -4 (-2,-1) 4 (2,-2) -8(1,0) 0nullwith(DEtools):with(plots):
wffc:=diff(y(x),x)=2*x*y(x):dsolve(wffc);
fangxiangcang:=DEplot(wffc,y(x),x=-2..2,y=-2..2,thickness=2):
jifenquxian:=contourplot(y/exp(x^2),x=-2..2,y=-2..2,contours=20,color=blue,thickness=2):
display(fangxiangcang);The slope fieldPoint Slope (1,1) 2 (-1,2) -4 (-2,-1) 4 (2,-2) -8(1,0) 0nullwith(DEtools):with(plots):
wffc:=diff(y(x),x)=2*x*y(x):dsolve(wffc);
fangxiangcang:=DEplot(wffc,y(x),x=-2..2,y=-2..2,thickness=2):
jifenquxian:=contourplot(y/exp(x^2),x=-2..2,y=-2..2,contours=20,color=blue,thickness=2):
display(fangxiangcang,jifenquxian);The slope field and solution curvesnullExample The slope field shown below is for the differential equationwith(DEtools):with(plots):
wffc:=diff(y(x),x)=y(x)+x:dsolve(wffc);
fangxiangcang:=DEplot(wffc,y(x),x=-2.5..2.5,y=-2.5..2.5,color=blue):
display(fangxiangcang);nullThe slope field shown below is for the differential equationThe slope depends on x only.On each vertical line the slope should be the same.But this is not the case.The same argument applies to (B) and (C).nullThe slope field shown below is for the differential equationThe slope should be positive in the third quadrant.But this is not the case.Thus (D) is the choice.nullLet’s choose some points to check the answer.Point Slope (1,1) 2 (-1,2) 1 (-1,-1) -2 (1,-1) 0(1,0) 1nullwith(DEtools):with(plots):
wffc:=diff(y(x),x)=x+y(x):dsolve(wffc);
fangxiangcang:=DEplot(wffc,y(x),x=-2.5..2.5,y=-2.5..2.5,color=blue):
jifenquxian:=contourplot((y+1+x)/exp(x),x=-2.5..2.5,y=-2.5..2.5,contours=30,thickness=2,color=red):
display(fangxiangcang,jifenquxian);The slope field and solution curvesnullwith(DEtools):with(plots):
wffc:=diff(y(x),x)=x+y(x):dsolve(wffc);
fangxiangcang:=DEplot(wffc,y(x),x=-.5..2.5,y=-.5..2.5,color=blue):
jifenquxian:=contourplot((y+1+x)/exp(x),x=-.5..2.5,y=-.5..2.5,contours=30,thickness=2,color=red):
display(fangxiangcang,jifenquxian);The slope field and solution curves
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