1

field=VectorPlot[{1,-4/49(T-21)},{t,0,20},{T,0,50},VectorScale->{Tiny,Tiny,None}]

3

VectorPlot[{1,y-1},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,y-x},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,(y^2)-(x^2)},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,(y^3)-(x^3)},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

4

VectorPlot[{1,y+x*Exp[-x]+1},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,2x*y+2x *Exp[-(x^2)]},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,-(2x+y)/(2y)},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,2 Sin[3x]+1+y},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

5

VectorPlot[{1,x+2y},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,y(4-y)},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,y^3/6-y-x^3/6},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,2x-1-y^2},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

6

Clear[f,t,y]
f[t_,y_]=-2+5t-4y;

h=0.5;
t[0]=0;
y[0]=-1.0;
t[i_]:=t[i]=t[i-1]+h
y[i_]:=y[i]=y[i-1]+h*f[t[i-1],y[i-1]]

euler=Table[{t[k],y[k]},{k,0,30}]
eulerplot=ListPlot[euler,PlotStyle->{PointSize[0.02],Red},AxesLabel->{"t","y"}]

y[0]
y[1]
y[2]
y[3]
y[4]

Clear[f,t,y]
f[t_,y_]=-2+5t-4y;

h=0.005;
t[0]=0;
y[0]=-1.0;
t[i_]:=t[i]=t[i-1]+h
y[i_]:=y[i]=y[i-1]+h*f[t[i-1],y[i-1]]

euler=Table[{t[k],y[k]},{k,0,300}];
eulerplot=ListPlot[euler,Joined->True,AxesLabel->{"t","y"}]

7

Solve[-2==k(70-20)]

field=VectorPlot[{1,-1/25(T-20)},{t,0,20},{T,0,50},VectorScale->{Tiny,Tiny,None}]

Troom=20.0;
TCoffee=95.0;
kdecay=-1/25;

Clear[f,t,y]
f[t_,y_]=kdecay(y-Troom);

h=2.0;
t[0]=0;
y[0]=TCoffee;
t[i_]:=t[i]=t[i-1]+h
y[i_]:=y[i]=y[i-1]+h*f[t[i-1],y[i-1]]

euler=Table[{t[k],y[k]},{k,0,30}]
eulerplot=ListPlot[euler,PlotStyle->{PointSize[0.02],Red},AxesLabel->{"t","y"}]

y[5]

Clear[f,t,y]
f[t_,y_]=-1/25(y-20);

h=0.01;
t[0]=0;
y[0]=95.0;
t[i_]:=t[i]=t[i-1]+h
y[i_]:=y[i]=y[i-1]+h*f[t[i-1],y[i-1]]

euler=Table[{t[k],y[k]},{k,0,6000}];
eulerplot2=ListPlot[euler,Joined->True,AxesLabel->{"t","y"}];

Show[eulerplot,eulerplot2]

8

VectorPlot[{1,-y(5-y)},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,y(4-y)},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,y^3/6-y-x^3/6},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]

VectorPlot[{1,2x-1-y^2},{x,-5,5},{y,-5,5},VectorScale->{Tiny,Tiny,None}]