Assignment 1 - FW00

> with(DEtools):

Question 1(c) (i)

> DE1:=diff(y(x),x)=3*x^4;

> DEplot(DE1,y(x),x=-2..2,[[y(2)=5],[y(2)=40],[y(2)=10],[y(2)=20],[y(2)=30],[y(2)=0]],y=-40..40,colour=black,linecolour=black);

Question 1(c) (ii)

> DE2:=diff(y(x),x)=2*exp(-2*x);

> DEplot(DE2,y(x),x=-1..3,[[y(0)=2],[y(0)=0],[y(0)=-2],[y(0)=5],[y(0)=-5],[y(0)=10]],y=-12..12,colour=black,linecolour=black);

Question 1(c) (iii)

> DE3:=diff(y(x),x)=tan(x);

> DEplot(DE3,y(x),x=-1.55..1.55,[[y(0)=2],[y(0)=0],[y(0)=1],[y(0)=-2],[y(0)=5],[y(0)=-5]],y=-5..10,stepsize=0.01,colour=black,linecolour=black);

Question 1(c) (iv)

> DE4:=diff(y(x),x)=1/(1+x)^2;

> DEplot(DE4,y(x),x=-3..3,[[y(1)=1],[y(1)=4],[y(1)=-2],[y(1)=5],[y(-2)=-5],[y(-2)=-2],[y(-2)=0]],y=-10..6,stepsize=0.01,colour=black,linecolour=black);

>

Question 2(c) (i)

> DE2a:=diff(y(x),x)= 1-y;

> DEplot(DE2a,y(x),x=-3..2,[[y(0)=5],[y(0)=1],[y(0)=-2],[y(0)=0],[y(0)=-5],[y(0)=3]],y=-10..10,stepsize=0.01,colour=black,linecolour=black);

Question 2(c) (ii)

> DE2b:=diff(y(x),x)=2*y+1;

> DEplot(DE2b,y(x),x=-2..2,[[y(0)=5],[y(0)=1],[y(0)=-2],[y(0)=0],[y(0)=-0.5],[y(0)=-5]],y=-10..10,stepsize=0.01,colour=black,linecolour=black);

Question 2(c) (iii)

> DE2c:=diff(y(x),x)=2*y^4;

> DEplot(DE2c,y(x),x=-3..3,[[y(1)=3],[y(1)=0],[y(-1)=-3],[y(1)=1/2],[y(-1)=-1/2]],y=-5..5,stepsize=0.005,colour=black,linecolour=black);

Question 2(c) (iv)

> DE2d:=diff(y(x),x)=y^(-3);

> DEplot(DE2d,y(x),x=-2..20,[[y(-2)=-1],[y(-2)=1],[y(-2)=-2],[y(-2)=2],[y(-2)=-3],[y(-2)=3]],y=-4..4,stepsize=0.01,colour=black,linecolour=black);

Question 3 (b)

> DE5:=diff(y(t),t)= a*y - c*y*y -h;

> a:=1;c:=1/10;h:=8/5;

> DEplot(DE5,y(t),t=0..10,[[y(0)=0],[y(0)=12],[y(0)=1],[y(0)=2],[y(0)=4],[y(0)=6],[y(0)=8],[y(0)=10]],y=-4..12,stepsize=0.1,colour=black,linecolour=black);

> h:=5/2;

> DEplot(DE5,y(t),t=0..10,[[y(0)=0],[y(0)=12],[y(0)=5],[y(0)=2],[y(0)=4],[y(0)=6],[y(0)=8],[y(0)=10]],y=-4..12,stepsize=0.1,colour=black,linecolour=black);

Question 3 (c)

> h:=3;

> DEplot(DE5,y(t),t=0..10,[[y(0)=0],[y(0)=12],[y(0)=1],[y(0)=2],[y(0)=4],[y(0)=6],[y(0)=8],[y(0)=10]],y=-4..12,stepsize=0.1,colour=black,linecolour=black);

From the graph we can see that at this level of harvesting, the fish population will die out no matter how large it is initially.