{VERSION 5 0 "IBM INTEL NT" "5.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "# Solution of homogeneous \+
first order equations\n\n# filename = 02.3.mws\n" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 9 "restart;\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 116 "# Example 2: Solution of first order homogeneous equ
ations via genhomosol\ndiffeqn:=x^2-y(x)^2+x*y(x)*diff(y(x),x)=0:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(DEtools):\nodeadvisor(d
iffeqn);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "gensol:=dsolve(
diffeqn,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "implicits
ol:=dsolve(diffeqn,y(x),implicit);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "tograph:=subs(y(x)=y,solve(implicitsol,_C1));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "contourvals:=[seq(subs(\{x=i
,y=0\},tograph),i=1..10)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
98 "with(plots):\ncontourplot(tograph,x=0.1..10,y=-5..5,contours=conto
urvals,grid=[60,60],color=BLACK);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
9 "Example 3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "M:=(x,y)->x
^(1/3)*y^(2/3)+x:\n  N:=(x,y)->x^(2/3)*y^(1/3)+y:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 43 "radsimp(M(t*x,t*y));\n  radsimp(N(t*x,t*y));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "odeadvisor(M(x,y(x))+N(
x,y(x))*diff(y(x),x)=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 
"dalembertsol(M(x,y(x))+N(x,y(x))*diff(y(x),x)=0);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 48 "dsolve(M(x,y(x))+N(x,y(x))*diff(y(x),x)=0,y
(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "dsolve(M(x,y(x))+N
(x,y(x))*diff(y(x),x)=0,y(x),implicit);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(DEtools);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 11 "?genhomosol" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "assu
me(x>0,y>0):\ngenhomosol(M(x,y(x))+N(x,y(x))*diff(y(x),x)=0,y(x));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "x:='x':y:='y':\nx:=v*y;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "D(x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 43 "step_1:=radsimp(M(x,y)*D(x)+N(x,y)*D(y)=0);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "step_2:=collect(step_1,\{D
(v),D(y)\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "step_3:=sub
s(\{D(y)=diff(y(v),v),y=y(v),D(v)=1\},step_2);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 34 "step_4:=separablesol(step_3,y(v));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "step_3:=subs(\{D(v)=diff(v(y),y),v=
v(y),D(y)=1\},step_2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "s
tep_4:=separablesol(step_3,v(y));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "x:='x':y:='y':v:='v':\nstep_5:=subs(\{y(v)=y,v=x/y\},
step_4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "assume(x>0,y>0)
:\nstep_6:=combine(simplify(step_5),ln);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "M(x,y(x))+N(x,y(x))*diff(y(x),x)=0;" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 31 "Application:  Models of Pursuit" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "simplify((-b*y/sqrt(x^2+y^2)+w)/(-b
*x/sqrt(x^2+y^2)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with
(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "gensol:=genho
mosol(diff(y(x),x)=(b*y(x)-w*sqrt(x^2+y(x)^2))/(b*x),y(x));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "x:='x':y:='y':\nb:='b':w:='w
':\ny:=u*x:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "step_1:=collect(radsimp(D(y)=(b*y-w
*sqrt(x^2+y^2))/(b*x)),\{D(x),D(u)\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "expand(step_1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "step_2:=subs(\{D(x)=diff(x(u),u),x=x(u),D(u)=1\},step
_1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "left_int:=int(1/sqr
t(1+u^2),u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "left_int:=c
onvert(left_int,ln);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "rig
ht_int:=int(-w/(b*x),x)+c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
59 "cval:=eval(solve(subs(\{x=1000,u=0\},left_int=right_int),c));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "step_3:=subs(c=cval,left_int
=right_int);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "step_4:=sol
ve(step_3,u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "y:='y':\ny
:=x*step_4;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "b:=1:\nwvals
:=seq(0.25*i,i=1..8):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "to
plot:=\{seq(subs(w=i,y),i=wvals)\};" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "plot(toplot,x=0..1000,color=BLACK,view=[0..1000,0..10
00]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 0 0
" 74 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }