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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# Orthogonal traject
ories\n\n# filename = 03.1.mws\n\nrestart;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 10 "\nExample 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 
"y[1]:=x->x:\n  y[2]:=x->sqrt(1-x^2):\n  D(y[1])(x);\n  D(y[2])(x);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "D(y[2])(sqrt(2)/2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(\{y[1](x),y[2](x),-x+sq
rt(2)\},x=-1..1,-.5..1.5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Exam
ple 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "x:='x':y:='y':\n  \+
sol:=dsolve(diff(y(x),x)=-x/(2*y(x)),y(x));" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 63 "implicitsol:=dsolve(diff(y(x),x)=-x/(2*y(x)),\n
\011\011y(x),implicit);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "
with(plots):\nc_vals:=seq(-1+i/4,i=0..8):\n  parabs:=\{seq(y=c*x^2,c=c
_vals)\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "IP_1:=implicit
plot(parabs,x=-3..3,y=-3..3,\n\011\011color=BLACK):\n  display(\{IP_1
\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "ellipses:=\{seq(y^
2+x^2/2=k,k=1..8)\}:\n  IP_2:=implicitplot(ellipses,x=-3..3,y=-3..3,co
lor=GRAY):\n  display(\{IP_2\});\n  display(\{IP_1,IP_2\});" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Example 3" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "Eq_1:=y^2+2*x*y-x^2=c:\n  step_1:=D(Eq_1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "step_2:=subs(\{D(x)=1,D(c)=0
\},step_1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "im_deriv:=so
lve(step_2,D(y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "step_3
:=simplify(-1/im_deriv);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 
"Diff_Eq:=diff(y(x),x)=subs(y=y(x),step_3);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 45 "with(DEtools):\n  Sol:=exactsol(Diff_Eq,y(x));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Eq_2:=subs(\{y(x)=y,_C[1]=k
\},Sol[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 267 "set_1:=\{se
q(subs(c=i,Eq_1),i=-3..3)\}:\n  set_2:=\{seq(subs(k=i,Eq_2),i=-3..3)\}
:\n  with(plots):\n  IP_1:=implicitplot(set_1,x=-4..4,y=-4..4,color=BL
ACK):\n  display(\{IP_1\});\n  IP_2:=implicitplot(set_2,x=-4..4,y=-4..
4,color=GRAY):\n  display(\{IP_2\});\n  display(\{IP_1,IP_2\});" }}}
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