pag. 22, #10(d). plotting the direction field
restart:
diffeq:=diff(y(x),x)=x^2+2*y(x)^2;
with(DEtools):
DEplot(diffeq,y(x),x=-1..0.4,[[0,0],[0,0.5],[0,1]],linecolor=black);
Look at the form of the direction field. Due to the fact that the direction field is rapidly increasind as x and y increase in modulus, we are restricting the range of the x on the right in order to avoid computational errors.
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pag. 28, #6. Euler method of the "broken--line": computing a sequence of points
restart:
Note that for the numeric option, no variables are allowed. If variables remain in the initial value problem, an error message will be issued that mentions Boolean variables.
diffeq:=diff(y(x),x)=x-y(x)^2;
inits:=y(1)=0;
Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5RzYiNiNJInhHRilGKywmRisiIiIqJClGJyIiI0YtISIi
Ly1JInlHNiI2IyIiIiIiIQ==
sol:=dsolve({diffeq,inits},y(x),numeric,method=classical[foreuler],stepsize=0.1);
Zio2I0kseF9jbGFzc2ljYWxHNiI2KUkkcmVzR0YlSSVkYXRhR0YlSSV2YXJzR0YlSSlzb2xucHJvY0dGJUkpb3V0cG9pbnRHRiVJJm5kc29sR0YlSSJpR0YlNiNJaW5Db3B5cmlnaHR+KGMpfjIwMDB+Ynl+V2F0ZXJsb29+TWFwbGV+SW5jLn5BbGx+cmlnaHRzfnJlc2VydmVkLkdGJUYlQyo+SThfRW52RFNOdW1lcmljU2F2ZURpZ2l0c0dGJUknRGlnaXRzR0YlPkYzIiM6QCUvSS1fRW52SW5Gc29sdmVHRiVJJXRydWVHJSpwcm90ZWN0ZWRHPkYrLSZJJmV2YWxmR0Y6NiNGMkYjPkYrLUY+RiM+RigtSSVldmFsR0Y6NiMmSTxkc29sdmUvbnVtZXJpYy9kYXRhL21vZHVsZXNHSShfc3lzbGliR0YlNiMiIiI+RiktX0YoSSRHZXRHRiU2I1Eoc3lzdmFyc0YlPkYqLUZNNiNRL3NvbG5fcHJvY2VkdXJlRiVAJDQtSSV0eXBlR0Y6NiRGKy5JKG51bWVyaWNHRjpDJEAsLUknbWVtYmVyR0Y6NiRGJDcrUSZzdGFydEYlLkkmc3RhcnRHRiVRJ21ldGhvZEYlLkknbWV0aG9kR0YlUSVsZWZ0RiVRJnJpZ2h0RiVRKWxlZnRkYXRhRiVRKnJpZ2h0ZGF0YUYlUStlbmdpbmVkYXRhRiVDJD5GJy1GKjYjLUkoY29udmVydEdGOjYkRiQuSSdzdHJpbmdHRjpAJi1GWDYkRicuSSZhcnJheUdGOk8tRkQ2JEYnRkowRidRKXByb2NuYW1lRiVPRictRmluNiRGJDcmUSVsYXN0RiUuSSVsYXN0R0YlUShpbml0aWFsRiUuSShpbml0aWFsR0YlQyRGaG9AJC1GWDYkRicuSSVsaXN0R0Y6TzcjLUkkc2VxR0Y6NiQvJkYpNiMsJkYtRkpGSkZKJkYnRmFyL0YtOyIiIUZKMy1GWDYkRitJIj1HRjotRmluNiQtSSRsaHNHRjo2I0YrNyRGYXFGYnFDJEAlLUZYNiQtSSRyaHNHRjpGX3NGaHE+RictRio2Iy9GYXE3JEZmci1JI29wR0Y6NiNGZXNZUUtpbml0aWFsfnZhbHVlc35tdXN0fmJlfnNwZWNpZmllZH5pbn5hfmxpc3RGJUZlcS9GK1Euc29sbnByb2NlZHVyZUYlTy1GRDYjRiovRitGUE9GKUAlMCUpcHJvY25hbWVHSSh1bmtub3duR0YlTy0uRmp0RiNDJD5GLC1JKHBvaW50dG9HRjo2IyYtRk02I1Ewc29sbl9wcm9jZWR1cmVzRiU2I0Zmck8tLkYsRiNaJUMkPkYnLUYqRl9zRltyRiVZRiVGJUYlRiU=
If you want a list of points, you can simply evaluate the function at the x values that you desire.
sol(1.1),sol(1.2),sol(1.3),sol(1.4),sol(1.5);
Nic3JC9JInhHNiIkIiM2ISIiLy1JInlHRiY2I0YlJCIzKTMrKysrKysrIiEjPTckL0YlJCIjN0YpL0YrJCIzXykqKioqKioqKioqKiozI0YwNyQvRiUkIiM4RikvRiskIjNqKSoqKioqKioqPWpDJEYwNyQvRiUkIiM5RikvRiskIjNHJ1EtJkhKJDRXJUYwNyQvRiUkIiM6RikvRiskIjNCb18qKWVVclZjRjA=
We also use some fancy Maple to form a sequence of output points. Note the use of the single forward quotes, which are a delayed evaluation operator. They allow the variable p to be replaced by a number before the numeric routine sol is executed. Failure to use the delayed evaluation quotes will yield a Boolean variable error as sol tries to work with a variable, p, rather than a number.
'sol(1+p*0.1)'$p=1..5;
(Remember: the expression before the $ must have delayed evaluation single forward quotes to avoid getting an error message about Booleans - i.e., trying to use a numerical routine on a variable.)
Nic3JC9JInhHNiIkIiM2ISIiLy1JInlHRiY2I0YlJCIzKTMrKysrKysrIiEjPTckL0YlJCIjN0YpL0YrJCIzXykqKioqKioqKioqKiozI0YwNyQvRiUkIiM4RikvRiskIjNqKSoqKioqKioqPWpDJEYwNyQvRiUkIiM5RikvRiskIjNHJ1EtJkhKJDRXJUYwNyQvRiUkIiM6RikvRiskIjNCb18qKWVVclZjRjA=
pag. 28, #7. Euler method of the "broken--line": computing the value in a fixed point after a sequence of steps
restart:
diffeq:=diff(x(t),t)=1+t*sin(t*x(t));
inits:=x(0)=0;
Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ4RzYiNiNJInRHRilGKywmIiIiRi0qJkYrRi0tSSRzaW5HNiRGJUkoX3N5c2xpYkdGKTYjKiZGK0YtRidGLUYtRi0=
Ly1JInhHNiI2IyIiIUYn
sol1:=dsolve({diffeq,inits},x(t),numeric,method=classical[foreuler],stepsize=evalf(1)):
sol1(evalf(1));
NyQvSSJ0RzYiJCIiIiIiIS8tSSJ4R0YlNiNGJEYm
sol2:=dsolve({diffeq,inits},x(t),numeric,method=classical[foreuler],stepsize=evalf(1)/2):
sol2(evalf(1));
NyQvSSJ0RzYiJCIiIiIiIS8tSSJ4R0YlNiNGJCQiMyEzajgpKik0Jj0xIiEjPA==
sol4:=dsolve({diffeq,inits},x(t),numeric,method=classical[foreuler],stepsize=evalf(1)/4):
sol4(evalf(1));
NyQvSSJ0RzYiJCIiIiIiIS8tSSJ4R0YlNiNGJCQiM10yKGVNMSc+UjYhIzw=
sol8:=dsolve({diffeq,inits},x(t),numeric,method=classical[foreuler],stepsize=evalf(1)/8):
sol8(evalf(1));
NyQvSSJ0RzYiJCIiIiIiIS8tSSJ4R0YlNiNGJCQiMz9MJ0doL3M6PiIhIzw=
pag. 28, #10. Euler method of the "broken--line": plotting Euler method output with specified stepsize, comparison with the true solution
Ordinarily, the output from dsolve/numeric is plotted using the odeplot command of the plots package. If you are content with this form of output, plotting Euler is easy. An example is given below.
If you are more particular and insist on seeing the kinks in the polygon, we have to work much harder. In this case, since we want to see the output with the given stepsize, we cannot use the odeplot command. The odeplot command picks values of x other than those which correspond to the stepsize. Instead, we need to make our own list of points, using only the x values at multiples of the stepsize.
(To expand on the previous comment on why one cannot use odeplot, note that the stepsize was indicated only in the dsolve/numeric command; odeplot knows nothing of this and uses a default of 49 points. The dsolve/numeric with given stepsize uses that stepsize, except perhaps at the last interval in the computation, where it uses a smaller step if the endpoint is not a multiple of the stepsize.This approach yields a smooth curve, rather than a polygonal one.)
Plot 1. Easy version.
restart:
diffeq:=diff(y(x),x)=x-y(x);
inits:=y(0)=0;
Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5RzYiNiNJInhHRilGKywmRisiIiJGJyEiIg==
Ly1JInlHNiI2IyIiIUYn
sol:=dsolve({diffeq,inits},y(x),numeric,
method=classical[foreuler],stepsize=0.1):
with(plots):
odeplot(sol,[x,y(x)],0..1);
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ptseq:='subs(sol(0+p*0.1),[x,y(x)])'$p=0..10;
Ni03JCQiIiFGJUYkNyQkIiIiISIiRiQ3JCQiIiNGKSQiMz8rKysrKysrNSEjPjckJCIiJEYpJCIzISkqKioqKioqKioqKioqKkdGLzckJCIiJUYpJCIzcioqKioqKioqKioqKjRjRi83JCQiIiZGKSQiMzUrKysrKytcISpGLzckJCIiJ0YpJCIzLSsrKysrVDk4ISM9NyQkIiIoRikkIjMzKysrKyFwSHkiRkQ3JCQiIilGKSQiM2grKysrQG4vQkZENyQkIiIqRikkIjNkKysrISpbP3VHRkQ3JCQiIzVGKSQiMyk0Kys1UyV5J1skRkQ=
plot({exp(-x)+x-1,[ptseq]},x=0..1,scaling=constrained);
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
To verify that the exact solution is y(x)=exp(-x)+x-1, we substitute:
subs(y(x)=exp(-x)+x-1,diffeq):
expand(%);
LywmKiQtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKiEiIkYtIiIiRi5GIw==
To verify that the initial condition is y(0)=0, we substitute:
subs(x=0,exp(-x)+x-1):
simplify(%);
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