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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "# Solution of separa
ble equations\n\n# filename = 02.2.mws\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "restart;\nwith(DEtools);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "?separablesol" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
43 "Example 1: solution of a separable equation" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 90 "y:='y':\nwith(student):\nstep1:=changevar(1-sq
rt(y)=u, Int(1/(2*y^(1/2)*(1-y^(1/2))),y), u);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "step2:=value(step1);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "step3:=subs(u=1-sqrt(y),step2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "step4:=int(1/x,x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "simplify(solve(step3=step4+c,x));\n
simplify(solve(step3=step4+c,y));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "gensol:=dsolve(diff(y(x),x)=(2*sqrt(y(x))-2*y(x))/x,y
(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "separablesol(diff(y(x),x)=(
2*sqrt(y(x))-2*y(x))/x,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "ivals:=seq(-2+0.25*i,i=0..16):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "toplot:=\{seq(subs(_C1=i,rhs(gensol)),i=ivals)\};" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot(toplot,x=-2..2,view=[-
2..2,0..4],color=BLACK);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Examp
le 2: solution of separable ode using each of dsolve and separablesol
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "sol1:=dsolve(diff(y(x),
x)=y(x)*cos(x)/(1+y(x)^2),y(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "sol2:=dsolve(D(y)(x)=y(x)*cos(x)/(1+y(x)^2),y(x),\n
\011\011implicit);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "odead
visor(D(y)(x)=y(x)*cos(x)/(1+y(x)^2));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "LHS:=int(cos(x),x);\n  RHS:=int((1+y^2)/y,y);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "with(DEtools):\n  sol2:=sepa
rablesol(\n\011\011D(y)(x)=y(x)*cos(x)/(1+y(x)^2),y(x));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "toplot:=subs(y(x)=y,lhs(sol2[1]));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "with(plots):\n  contour
plot(toplot,x=0..10,y=0..10,\n\011\011color=BLACK,contours=10,grid=[60
,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eval(subs(\{x=0,y
=1\},toplot));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Application:  s
olution of a kidney dialysis separable ode" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 115 "alpha:='alpha':\nQ:='Q':u0:='u0':k:='k':\nu:='u':v
:='v':\ny:='y':L:='L':\nstep1:=dsolve(diff(y(x),x)=-alpha*y(x),y(x));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "y:=subs(_C1=c,rhs(step1
));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "step2:=dsolve(\{diff
(u(x),x)=-k/Q[b]*y,u(0)=u0\},u(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "leftside:=subs(x=L,rhs(step2)-y);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 27 "\ncval:=solve(leftside=0,c);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "u:=simplify(subs(c=cval,rhs(step2))
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(x=L,u);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v:=subs(c=cval,u-y);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "alpha:=k/Q[b]-k/Q[d];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "k:=2.25:\nL:=1:\nQ[b]:=2:\nQ
[d]:=4:\nu0:=34+1.8:\nu;\nv;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "plot(u,x=0..1);\nplot(v,x=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "plot(14.2623*(-1+exp(0.5625*(1-x))),x=0..1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 }
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