{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {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 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "#sfb Nov 6, 2001" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# 10.1 What is a differenti al equation? (See also 6.3)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "#(The horizontal axis is often t, the vertical axis is y.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "print(dy/dt = F(y,t));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "dy/dt = F(y,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "#the above is a First Order Example " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "print (d^2*y/dt^2 = G(d y/dt,y,t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "#the above i s a Second Order Example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " #" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "#In 6.3 you solved DE' s of the form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "print(dy/ dx = f(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "#And Initial value problems like" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "pri nt(dy/dx = f(x), y(a)= b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "#Solving DE's has one big difference from solving poly's" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "#0. Solutions of polys are n umbers, Solutions to DE's are functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "#Solvi ng DE's has many features in common with finding roots of polys" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#1. It is easier to check th en to solve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "#2. The num ber of solutions increases with degree (resp. order)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "#3. Sometimes the solutions can on ly be found numerically or graphically. The is no general solution tha t always works." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "#4. Some times only knowing properties of solutions will solve the mathematical problem in question." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "#5 . There is much more to either, than just solving them." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# On the otherhand solving DE's has new features not shared by polys" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "#6. There are always infinitely many solutions due t o constants of integration. First order equations have one arbitrary c onstant and Second order equations have two arbitrary constants." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "#7. Instead of solving a DE , the problem is often solving a DE with additional conditions (often \+ initial) which `eliminate' the constants in #5. That is you solve the \+ INITIAL VALUE PROBLEM which includes the DE and initial values:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "#IVP1: y'=F(y,t) and y(0)=y_0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "#IVP2: y''=G(y',y,t) and y(0)=y_0 and y'(0)=y _1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 63 "# checking is easy -- show y=100sin(t) is a \+ solution to y''+y=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "y:=1 00*sin(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "yprimeprime:= diff(y,t,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "yprimeprime +y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "#show y=(e^x+e^-x)/2 is a solution to y''=sqrt(1+(y')^2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "y:=(exp(x)+exp(-x))/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "yprime:=diff(y,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "yprimeprime:=diff(y,x,x);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "yprimeprime-sqrt(1+yprime^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "expand(1+yprime^2);expand(y^2);#HVC" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "expand(1+yprime^2)-expand(y^ 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "yprimeprime-y;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "#Variation on checking is ea sy -- Find which k's is y=e^(kt) is a solution y'=5y" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "y:=exp(k*t);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "yprime:=diff(y,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eqn:=yprime=5*y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(eqn,k);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "lhs(eqn)/y=rhs(eqn)/y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# slope fields y' = F(y,x)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "# y'=y(4-y)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "with(DEtools);y :='y';dfieldplot(diff(y(x),x)=y(x)*(4-y(x)),y(x),x=0..3,y=-1..5,arrows =LARGE);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 75 "y:='y';a:=dfieldplot(diff(y(x),x)=y(x),y(x),x= -3..3,y=-3..3,arrows=LINE):a;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "b:=plot([-2*exp(x),-exp(x),0*exp(x),exp(x),2*exp(x)],x=-3..3,y=- 3..3,color=blue):b;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with (plots);display(a,b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "a2 :=dfieldplot(diff(y(x),x)=y(x),y(x),x=-3..3,y=-3..3,arrows=LARGE):disp lay(a2,b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "b2:=plot([-e xp(x),-0.5*exp(x),-0.25*exp(x),0.25*exp(x),0.5*exp(x),exp(x)],x=-3..3, y=-3..3,color=blue):b;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "d isplay(a,b2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fieldplot([1,x/y],x=-3..3,y= -3..3,arrows=LINE);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "dfie ldplot(diff(y(x),x)=x/y(x),y(x),x=-3..3,y=-3..3,arrows=LARGE);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 62 "# The big three: Existence, Uniqueness, Continuity \+ w/resp Data" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "#1. Is there always a solution to IVP y'=F(y,x) y(0)=y_0? Yes if F is continuous. \+ (Existence)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "#2. Is the \+ solution to IVP y'=F(y,x) y(0)=y_0 unique? Yes if slightly more than F continuous is true. This means solution curves cannot cross. (Uniquen ess)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "#3. Do small chang es in y_0 give small changes in the solution? (Continuity w/resp Data) . Not always true, the butterfly effect." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "# From a later lecture" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "# Maple does DE's" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ode:=diff(y(t),t)=k*y(t);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "dsolve(ode);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ode2:=diff(y(t),t)=y(t)*(4-y(t));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "dsolve(ode2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sol:=dsolve(\{ode2,y(0)=0.5\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A:=plot(rhs(sol),t=0..5,color=blue):A;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "B:=dfieldplot(ode2,y(t),t=0..5,y=-1 ..5,arrows=LINE):B;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with (plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "display(A,B);" }}}}{MARK "68 0 0" 22 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }