reduce-historical/r36/xlog/ODESOLVE.LOG

REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...


% Tests and demonstrations for the odesolve package

% First some tests of the testdf module
algebraic procedure showode();
 <<write "order is ", odeorder, " and degree is ", odedegree;
   write "linearity is ", odelinearity," and highestderiv is ",
         highestderiv>>;


showode


depend y,x$


ode1 := df(y,x);


ode1 := df(y,x)

sortoutode(ode1, y, x)$


showode()$


order is 1 and degree is 1

linearity is 1 and highestderiv is df(y,x)

sortoutode(ode1**2,y,x)$


showode() $


order is 1 and degree is 2

linearity is 2 and highestderiv is df(y,x)

sortoutode(e**ode1,y,x) $


showode() $


order is 1 and degree is 0

              df(y,x)
linearity is e        and highestderiv is df(y,x)

sortoutode(df(y,x)*df(y,x,2),y,x) $


showode() $


order is 2 and degree is 1

linearity is 2 and highestderiv is df(y,x,2)

nodepend y,x $


depend z,w $


sortoutode(df(z,w,2)+3*z*df(z,w)+e**z,z,w) $


showode() $


order is 2 and degree is 1

              z
linearity is e  and highestderiv is df(z,w,2)

nodepend z,w $



% ******************************************
% Next some tests for first-order differential equations
depend y,x $


% Just to test tracing
on trode $



% First-order quadrature case
ode := df(y,x) - x**2 - e**x;


                  x    2
ode := df(y,x) - e  - x

odesolve(ode, y, x);

This first-order ODE can be solved by quadrature
                       x    3
    3*arbconst(1) + 3*e  + x
{y=---------------------------}
                3


% A first-order linear equation, with an initial condition
ode:=df(y,x) + y * sin x/cos x - 1/cos x;


        cos(x)*df(y,x) + sin(x)*y - 1
ode := -------------------------------
                   cos(x)

ans:=odesolve(ode,y,x);

This is a first-order linear ODE solved by the integrating factor method
ans := {y=arbconst(2)*cos(x) + sin(x)}

% Note that arbconst is declared as an operator
% The initial condition is y = 1 at x = 0 so we do...
arbconst(!!arbconst)
   := sub(y=1,x=0,rhs first solve(ans,arbconst(!!arbconst)));


arbconst(2) := 1

ans;


{y=cos(x) + sin(x)}

clear arbconst(!!arbconst) $



% A simple separable case
ans := odesolve(df(y,x) - y**2,y,x);

This is a first-order separable ODE
         arbconst(3)*y - x*y - 1
ans := {-------------------------=0}
                    y

% We can improve this by
solve(ans,y);


           1
{y=-----------------}
    arbconst(3) - x

nodepend y,x $



% A separable case, in different variables, with an initial condition
depend z,w $


ode:= (1-z**2)*w*df(z,w)+(1+w**2)*z;


                     2                2
ode :=  - df(z,w)*w*z  + df(z,w)*w + w *z + z

% Assign the answer so we can input the condition (z = 2 at w = 1/2)
ans:=odesolve(ode,z,w);

This is a first-order separable ODE
                                                2    2
         2*arbconst(4) - 2*log(w) - 2*log(z) - w  + z
ans := {-----------------------------------------------=0}
                               2

% To tidy up the answer we will get for the constant we use
for all x let log(x)+log(1/x)=0 $


arbconst(!!arbconst) := sub(z=2,w=1/2,
                            rhs first solve(ans,arbconst(!!arbconst)));


                 - 15
arbconst(4) := -------
                  8

ans;


                              2      2
   - 8*log(w) - 8*log(z) - 4*w  + 4*z  - 15
{-------------------------------------------=0}
                      8

clear arbconst(!!arbconst) $


nodepend z,w $



% Now a homogeneous one
depend y,x $


ode:=df(y,x) - (x-y)/(x+y);


        df(y,x)*x + df(y,x)*y - x + y
ode := -------------------------------
                    x + y

% To make this look decent...
for all x,w let e**((log x)/w)=x**(1/w),
                (sqrt w)*(sqrt x)=sqrt(w*x) $


ans := odesolve(ode,y,x);

This is a first-order ODE of algebraically homogeneous type
 solved by change of variables y = vx method
                               2            2
ans := {arbconst(5) + sqrt( - x  + 2*x*y + y )=0}


% Reducible to homogeneous 
% Note this is the previous example with origin shifted
ode:=df(y,x) - (x-y-3)/(x+y-1);


        df(y,x)*x + df(y,x)*y - df(y,x) - x + y + 3
ode := ---------------------------------------------
                         x + y - 1

ans := odesolve(ode,y,x);

This is a first-order ODE reducible to homogeneous type
solved by shifting the origin
                               2                  2
ans := {arbconst(6) + sqrt( - x  + 2*x*y + 6*x + y  - 2*y - 7)=0}


% and the special case of reducible to homogeneous
ode:=df(y,x)-(2*x+3*y+1)/(4*x+6*y+1);


        4*df(y,x)*x + 6*df(y,x)*y + df(y,x) - 2*x - 3*y - 1
ode := -----------------------------------------------------
                           4*x + 6*y + 1

ans := odesolve(ode,y,x);

This is a first-order ODE reducible to homogeneous type
belonging to the special case where top and bottomare parallel lines
solved by new variable and separation
         49*arbconst(7) - 3*log(14*x + 21*y + 5) - 21*x + 42*y
ans := {-------------------------------------------------------=0}
                                  49


% To tidy up the next one we need
for all x,w let e**(log x + w) = x*e**w,
                e**(w*log x)=x**w $



% a Bernoulli equation
ode:=x*(1-x**2)*df(y,x) + (2*x**2 -1)*y - x**3*y**3;


                   3                3  3      2
ode :=  - df(y,x)*x  + df(y,x)*x - x *y  + 2*x *y - y

odesolve(ode,y,x);

This is a first-order ODE of Bernoulli type
                          5
  1    5*arbconst(8) + 2*x
{----=----------------------}
   2          4      2
  y        5*x  - 5*x


% and finally, in this set, an exact case
ode:=(2*x**3 - 6*x*y + 6*x*y**2) + (-3*x**2 + 6*x**2*y - y**3)*df(y,x);


                  2                2            3      3        2
ode := 6*df(y,x)*x *y - 3*df(y,x)*x  - df(y,x)*y  + 2*x  + 6*x*y  - 6*x*y

odesolve(ode,y,x);

This is an exact first order ODE
                    4       2  2       2      4
{4*arbconst(9) + 2*x  + 12*x *y  - 12*x *y - y =0}


% ******************************************
% Now for higher-order linear equations with constant coefficients

% First, examples without driving terms
% A simple one to start
ode:=6*df(y,x,2)+df(y,x)-2*y;


ode := 6*df(y,x,2) + df(y,x) - 2*y

odesolve(ode,y,x);

This is a linear ODE with constant coefficients of order 2
                    (7*x)/6
    arbconst(11) + e       *arbconst(10)
{y=--------------------------------------}
                   (2*x)/3
                  e


% An example with repeated and complex roots
ode:=df(y,x,4)+2*df(y,x,2)+y;


ode := df(y,x,4) + 2*df(y,x,2) + y

odesolve(ode,y,x);

This is a linear ODE with constant coefficients of order 4
{y= - arbconst(15)*sin(x)*x + arbconst(14)*cos(x)*x - arbconst(13)*sin(x)

  + arbconst(12)*cos(x)}


% A simple right-hand-side using the above example;
% It will need the substitution
for all w let (sin w)**2 + (cos w)** 2 = 1 $


ode:=ode-exp(x);


                                  x
ode := df(y,x,4) + 2*df(y,x,2) - e  + y

odesolve(ode,y,x);

This is a linear ODE with constant coefficients of order 4
{y=( - 4*arbconst(19)*sin(x)*x + 4*arbconst(18)*cos(x)*x - 4*arbconst(17)*sin(x)

                                x
     + 4*arbconst(16)*cos(x) + e )/4}


ode:=df(y,x,2)+4*df(y,x)+4*y-x*exp(x);


                                x
ode := df(y,x,2) + 4*df(y,x) - e *x + 4*y

ans:=odesolve(ode,y,x);

This is a linear ODE with constant coefficients of order 2
                                                    3*x        3*x
           27*arbconst(21)*x + 27*arbconst(20) + 3*e   *x - 2*e
ans := {y=---------------------------------------------------------}
                                       2*x
                                   27*e

% At x=1 let y=0 and df(y,x)=1
ans2 := solve({first ans, 1 = df(rhs first ans, x)}, 
               {arbconst(!!arbconst-1),arbconst(!!arbconst)});


                         2*x     x  2      x        x
                        e   *(9*e *x  - 6*e *x + 2*e  - 54*x*y - 27*x + 27*y)
ans2 := {{arbconst(20)=-------------------------------------------------------,
                                                 27

                         2*x        x      x
                        e   *( - 3*e *x + e  + 18*y + 9)
          arbconst(21)=----------------------------------}}
                                       9

arbconst(!!arbconst -1) := sub(x=1,y=0,rhs first first ans2);


                  2
                 e *(5*e - 27)
arbconst(20) := ---------------
                      27

arbconst(!!arbconst) := sub(x=1,y=0,rhs second first ans2);


                  2
                 e *( - 2*e + 9)
arbconst(21) := -----------------
                        9

ans;


       3*x        3*x      3        3       2         2
    3*e   *x - 2*e    - 6*e *x + 5*e  + 27*e *x - 27*e
{y=-----------------------------------------------------}
                              2*x
                          27*e

clear arbconst(!!arbconst),arbconst(!!arbconst-1), ans, ans2 $



% For simultaneous equations you can use the machine e.g. as follows

depend z,x $


ode1:=df(y,x,2)+5*y-4*z+36*cos(7*x);


ode1 := 36*cos(7*x) + df(y,x,2) + 5*y - 4*z

ode2:=y+df(z,x,2)-99*cos(7*x);


ode2 :=  - 99*cos(7*x) + df(z,x,2) + y

ode:=df(ode1,x,2)+4*ode2;


ode :=  - 2160*cos(7*x) + df(y,x,4) + 5*df(y,x,2) + 4*y

y := rhs first odesolve(ode,y,x);

This is a linear ODE with constant coefficients of order 4
y := arbconst(25)*sin(x) + arbconst(24)*cos(x) - arbconst(23)*sin(2*x)

      + arbconst(22)*cos(2*x) + cos(7*x)

z := rhs first solve(ode1,z);


z := (4*arbconst(25)*sin(x) + 4*arbconst(24)*cos(x) - arbconst(23)*sin(2*x)

       + arbconst(22)*cos(2*x) - 8*cos(7*x))/4

clear ode1, ode2, ode, y,z $


nodepend z,x $



% A "homogeneous" n-th order (Euler) equation
ode := x*df(y,x,2) + df(y, x) + y/x + (log x)**3;


                   2                     3
        df(y,x,2)*x  + df(y,x)*x + log(x) *x + y
ode := ------------------------------------------
                           x

odesolve(ode, y, x);

This equation is of the homogeneous (Euler) type
                                                                       3
{y=( - 2*arbconst(27)*sin(log(x)) + 2*arbconst(26)*cos(log(x)) - log(x) *x

               2
     + 3*log(x) *x - 3*log(x)*x)/2}


% Not yet working
% ode :=6*df(y,x,2)+df(y,x)-2*y + tan x;
% odesolve(ode, y,x);

% To reset the system
!!arbconst := 0 $


clear ode $


off trode$


nodepend y,x $


end $
(TIME:  odesolve 2950 3310)