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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "\n# verify solutions, solv
e and plot solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 23 "# filename = 01.2.mws\n\n" }{MPLTEXT 1 0 8 "restart;" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Example 1 -Verify the solution o
f an ode" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:=x->x*exp(-x)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "dy:=diff(y(x),x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "d2y:=diff(y(x),x$2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d2y+2*dy+y(x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(y(x),x=-1..1);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "Example 2: verify an implicit solution" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "x:='x':y:='y':\nstep_1:=D(
2*x^2+y^2-2*x*y+5*x=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "
step_2:=subs(D(x)=1,step_1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "solve(step_2,D(y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "implici
tplot(2*x^2+y^2-2*x*y+5*x=0,x=-7..2,y=-7..2);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 7 "?dsolve" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "E
xample 3: solve an ode using dsolve and plot multiple solutions" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "sol:=dsolve(diff(y(x),x)=-y(
x)*cos(x),y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "toplot:
=\{seq(subs(_C1=i,rhs(sol)),i=-5..5)\}:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "plot(toplot,x=0..4*Pi,color=BLACK);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 64 "Example 4: solve an ode using dsolve and plot m
ultiple solutions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sol:=d
solve(diff(y(x),x$2)+y(x)=0,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 133 "ivals:=seq(.5*i,i=-5..5):\ntoplot1:=\{seq(subs(\{_C1
=i,_C2=0\},rhs(sol)),i=ivals)\}:\ntoplot2:=\{seq(subs(\{_C1=0,_C2=i\},
rhs(sol)),i=ivals)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "A
:=array(1..2):\nA[1]:=plot(toplot1,x=0..4*Pi,color=BLACK):\nA[2]:=plot
(toplot2,x=0..4*Pi,color=BLACK):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "with(plots):\ndisplay(A);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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