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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "# Exact equations\n\n# fil
ename = 02.4.mws\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restar
t;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "# Example 2: Exact \+
equations\nwith(DEtools);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "diffeq:=2*x*sin(y(x))+(x^2*cos(y(x))-1)*diff(y(x),x)=0:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "odeadvisor(diffeq);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sol1:=exactsol(diffeq,y(x));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sol2:=dsolve(diffeq,y(x
));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "gensol:=lhs(subs(\{_
C1=0,y(x)=y\},sol2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "wi
th(plots):\ncontourplot(gensol,x=-5..5,y=-5..5,color=BLACK,grid=[70,70
]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "sol3:=dsolve(\{diffe
q,y(0)=1/2\},y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "step
1:=subs(\{y(x)=1/2,x=0\},sol2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "cval:=solve(step1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "tograph:=subs(\{_C1=cval,y(x)=y\},sol2);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 80 "with(plots):\nimplicitplot(tograph,x=-10..10,y=-
10..10,color=BLACK,grid=[70,70]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
39 "Example 3: test for exactness and solve" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 66 "capm:=(x,y)->2*x-y^2*sin(x*y):\ncapn:=(x,y)->cos(x
*y)-x*y*sin(x*y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "testeq
(diff(capm(x,y),y)=diff(capn(x,y),x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "f:=int(capm(x,y),x)+g(y);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "diff(f,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "solve(diff(f,y)=capn(x,y),diff(g(y),y));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "f:=subs(g(y)=0,f);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 73 "with(plots):\ncontourplot(f,x=0..3*Pi,y=0..3*Pi,gri
d=[70,70],color=BLACK);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "
step1:=dsolve(capm(x,y(x))+capn(x,y(x))*diff(y(x),x)=0,y(x),implicit);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "with(DEtools):\naltsol:
=exactsol(capm(x,y(x))+capn(x,y(x))*diff(y(x),x)=0,y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:=subs(\{_C1=0,y(x)=y\},lhs(step1)
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "with(linalg):\n  pote
ntial([capm(x,y),capn(x,y)],[x,y],'F');" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "F;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Example 4: \+
Solve exact ode with ic's" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
64 "y:='y':\ngensol:=dsolve(1+5*x-y(x)-(x+2*y(x))*diff(y(x),x),y(x));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "implicitsol:=dsolve(1+5
*x-y(x)-(x+2*y(x))*diff(y(x),x),y(x),implicit);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 63 "\013sol:=dsolve(\{1+5*x-y(x)-(x+2*y(x))*diff(y
(x),x),y(0)=1\},y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a
ssign(sol):\nplot(y(x),x=-5..5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 41 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
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