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bc2

bc2(%04,x=0,y=1,x=1,y...

Calculate

bc2-diff

u:'diff(y,x)^2+4*y*y+...

c11: bc2(sol, x=-1, y...

uy: diff(u,y);

Calculate

bc2-diff-ode2-plot2d-rhs
plot2d(rhs(eqnend),[x,-1,2],[xlabel,"x"],[ylabel,"y(x)"]);

eqn:diff(y,x,2)+y=0;

sol:ode2(eqn,y,x);

eqnend:bc2(sol, x=-1,...

Calculate

bc2-diff-ic2-ode2-ratsimp

eqn3:y*('diff(y,x))^3...

soln3:ode2(eqn3,y,x);

ratsimp(ic2(soln3,x=0...

Calculate

bc2-diff-ode2

eqn:'diff(y,x,2)-y=x*x;

sol:ode2(eqn,y,x);

bc2(sol, x=1, y=1,...

Calculate

bc2-diff-ode2

eqn: 'diff(y,x,2)-3*'...

sol: ode2(eqn,y,x);

sol1:bc2(sol, x=-2, y...

Calculate

bc2-diff-ode2-plot2d-rhs-sin-subst
plot2d(rhs(eq),[x,-1,1],[xlabel,"x"],[ylabel,"y(x)"]);

F:2*Dy^2+2*y^2-y*%e^(...

dF:diff(F,Dy);

eqn:diff(F,y)-diff(dF...

Calculate

bc2-diff-ode2-plot2d-rhs
          plot2d(rhs(eqnend),[x,-1,2],[xlabel,"x"],[ylabel,"y(x)"]);

eqn:diff(y,x,2)-y=x*x;

sol:ode2(eq...

eqnend:bc2(...

Calculate

bc2-cos-diff-ode2-plot2d-rhs
plot2d(rhs(sol1),[x,-4,0],[xlabel,"x"],[ylabel,"y(x)"]);

eqn4:diff(y(x),x,2)+d...

sol:ode2(eqn4,y,x);

sol1:bc2(sol,x=-4,y=1...

Calculate

bc2

Run Example
(%i1)kill(all);
(%o0)                                done
(%i1) depends(f,[x,y]);
(%o1)                              [f(x, y)]
(%i2) depends(y,x);
(%o2)                               [y(x)]
(%i3) f(x,y):=12*x*y+(diff(y,x))^2;
                                                2
(%o3)                   f(x, y) := 12 x y + diff (y, x)
(%i4) l1:diff(f(x,y),y);
(%o4)                                12 x
(%i5) l2:diff(f(x,y),'diff(y,x));
                                       dy
(%o5)                                2 --
                                       dx
(%i6) l3:diff(l2,x);
                                        2
                                       d y
(%o6)                                2 ---
                                         2
                                       dx
(%i7) y1:ode2(l1-l3=0,y,x);
                                  3
(%o7)                        y = x  + %k2 x + %k1
(%i8) bc2(%,x=0,y=3,x=1,y=6);
(%o8)                                 []
(%i9) 
Run Example
eqn:diff(y,x,2)+y=0;
(%o1)                                y = 0
(%i2)           sol:ode2(eqn,y,x);
(%t2)                                y = 0

                                     msg1

(%o2)                                false
(%i3)           eqnend:bc2(sol, x=-1, y=1, x=1, y=-2);
(%o3)                                 []
(%i4)           plot2d(rhs(eqnend),[x,-1,2],[xlabel,"x"],[ylabel,"y(x)"]);
plot          plot2d(rhs(eqnend),[x,-1,2],[xlabel,"x"],[ylabel,"y(x)"]);
Run Example
T:2*y1^2-y1*y2+3*y2^2+2*y2;
                             2                      2
(%o1)                    3 y2  - y1 y2 + 2 y2 + 2 y1
(%i2) F:2*Dy^2+2*y^2+y*%e^x+4*x*%e^2*x;
                           2     x         2  2       2
(%o2)                   2 y  + %e  y + 4 %e  x  + 2 Dy
(%i3) x1:-1;
(%o3)                                 - 1
(%i4) x2:1;
(%o4)                                  1
(%i5) dF:diff(F,Dy);
(%o5)                                4 Dy
(%i6) eqn:diff(F,y)-diff(dF,x)-diff(dF,y)*'diff(y,x)-diff(dF,Dy)*'diff(y,x,2);
                                   2
                                  d y           x
(%o6)                         - 4 --- + 4 y + %e
                                    2
                                  dx
(%i7) eqn:subst(Dy='diff(y,x),eqn);
                                   2
                                  d y           x
(%o7)                         - 4 --- + 4 y + %e
                                    2
                                  dx
(%i8) sol:ode2(eqn,y,x);
                                    x
                        (2 x - 1) %e          x         - x
(%o8)               y = ------------- + %k1 %e  + %k2 %e
                             16
(%i9) lab2:bc2(sol,x=x1,y=2,x=x2,y=0.5), numer;

rat: replaced 2.068977395219645 by 11998/5799 = 2.068977409898258

rat: replaced -0.3678794411714 by -8544/23225 = -0.3678794402583

rat: replaced -2.71828182845905 by -23225/8544 = -2.71828183520599

rat: replaced 0.33010738572131 by 2490/7543 = 0.33010738432984

rat: replaced -2.71828182845905 by -23225/8544 = -2.71828183520599

rat: replaced -0.3678794411714 by -8544/23225 = -0.3678794402583
                             x                       x                      - x
(%o9) y = 0.0625 (2 x - 1) %e  + 0.018775282064164 %e  + 0.75859329335714 %e
(%i10) linEq1: subst([x=x1,y=-1],sol);
                                                     - 1
                                       - 1       3 %e
(%o10)                - 1 = %e %k2 + %e    %k1 - -------
                                                   16
(%i11) linEq2: subst([y1=rhs(sol),Dy=rhs(diff(sol, x))],dF=0);
                              x               x
                  (2 x - 1) %e          x   %e          - x
(%o11)         4 (------------- + %k1 %e  + --- - %k2 %e   ) = 0
                       16                    8
(%i12) linEq2: subst([x=x2,linEq2]);
(%o12)                                - 1
(%i13) con: solve([linEq1, linEq2], [%k1,%k2]);
(%o13)                                []
(%i14) lab3:subst(con,sol);
                                    x
                        (2 x - 1) %e          x         - x
(%o14)              y = ------------- + %k1 %e  + %k2 %e
                             16
(%i15) K1:diff(T,y1);
(%o15)                             4 y1 - y2
(%i16) K2:diff(T,y2);
(%o16)                           6 y2 - y1 + 2
(%i17) dF=K1;
(%o17)                         4 Dy = 4 y1 - y2
(%i18) dF=-K2;
(%o18)                      4 Dy = - 6 y2 + y1 - 2
(%i19) linEq: subst([y=rhs(sol),Dy=rhs(diff(sol, x))],dF);
                                x               x
                    (2 x - 1) %e          x   %e          - x
(%o19)           4 (------------- + %k1 %e  + --- - %k2 %e   )
                         16                    8
(%i20) left1: subst([x=x1],linEq);
                                                   - 1
                                       - 1       %e
(%o20)                 4 (- %e %k2 + %e    %k1 - -----)
                                                  16
(%i21) right1: subst([y1=subst(x=x1,rhs(sol)),y2=subst(x=x2,rhs(sol))],K1);
                                      - 1
                        - 1       3 %e         - 1                %e
(%o21)    4 (%e %k2 + %e    %k1 - -------) - %e    %k2 - %e %k1 - --
                                    16                            16
(%i22) left2: subst([x=x2],linEq);
                               - 1                3 %e
(%o22)                  4 (- %e    %k2 + %e %k1 + ----)
                                                   16
(%i23) right2: subst([y2=subst(x=x1,rhs(sol)),y2=subst(x=x2,rhs(sol))],K2);
                                                     - 1
                                       - 1       3 %e
(%o23)            - y1 + 6 (%e %k2 + %e    %k1 - -------) + 2
                                                   16
(%i24) R1:left1=right1;
                                   - 1
                       - 1       %e
(%o24) 4 (- %e %k2 + %e    %k1 - -----) = 
                                  16
                                                 - 1
                                   - 1       3 %e         - 1                %e
                     4 (%e %k2 + %e    %k1 - -------) - %e    %k2 - %e %k1 - --
                                               16                            16
(%i25) R2:left2=-right2;
              - 1                3 %e
(%o25) 4 (- %e    %k2 + %e %k1 + ----) = 
                                  16
                                                                       - 1
                                                         - 1       3 %e
                                      y1 - 6 (%e %k2 + %e    %k1 - -------) - 2
                                                                     16
(%i26) con: solve([R1, R2], [%k1,%k2]);
                                    3                      4        2
                 %e (8 y1 - 16) + %e  (128 - 64 y1) + 51 %e  - 56 %e  - 7
(%o26) [[%k1 = - --------------------------------------------------------, 
                                        4         2
                                  304 %e  + 320 %e  - 48
                                               3                4       2
                                             %e  (4 - 2 y1) + %e  - 7 %e  - 6
                                     %k2 = - --------------------------------]]
                                                        4        2
                                                   76 %e  + 80 %e  - 12
(%i27) lab4:subst(con,sol);
               x                     3                      4        2
             %e  (%e (8 y1 - 16) + %e  (128 - 64 y1) + 51 %e  - 56 %e  - 7)
(%o27) y = - --------------------------------------------------------------
                                       4         2
                                 304 %e  + 320 %e  - 48
                         - x    3                4       2                    x
                       %e    (%e  (4 - 2 y1) + %e  - 7 %e  - 6)   (2 x - 1) %e
                     - ---------------------------------------- + -------------
                                      4        2                       16
                                 76 %e  + 80 %e  - 12
(%i28) plot2d([rhs(lab2), rhs(lab3), rhs(lab4)], [x,x1,x2], [xlabel,"x"], [ylabel,"y(x)"], [color, red, green, blue], [legend, "y(x)=lab2", "y(x)=lab3", "y(x)=lab4"]);
plotplot2d([rhs(lab2), rhs(lab3), rhs(lab4)], [x,x1,x2], [xlabel,"x"], [ylabel,"y(x)"], [color, red, green, blue], [legend, "y(x)=lab2", "y(x)=lab3", "y(x)=lab4"]);

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