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algexact-algsys-cos-false-sin-true

algexact:true;

algsys([x^3+(cos(%pi/...

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algexact-numer-solve-true

/* Brunnenproblem - f...

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numer:true;

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algexact:true;

algsys([x^3+(cos(alph...

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Y:16/10*b*b*8=60/1000;

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algexact:false;

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algexact-algsys-cos-false-sin-true

algexact:true;

algsys([x^3+(cos(alph...

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algexact-numer-solve-true

numer:true;

algexact:true;

y0:2;

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algexact-algsys-false

algexact:false;

eq1:eps^3+a*eps^2+b=0;

alpha=15;

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algexact-algsys-matrix-true

algexact:true;

eq1:0=I0-U1/R1-(U1-U2...

eq2:0=I0-U2/R2-(U2-U1...

Calculate

algexact

Run Example
(%i1)algexact:true;
(%o1)                                true
(%i2) eq1:0=I0-U1/R1-(U1-U2)/R12-(U1-U3)/R13;
                            U3 - U1   U2 - U1   U1
(%o2)                   0 = ------- + ------- - -- + I0
                              R13       R12     R1
(%i3) eq2:0=I0-U2/R2-(U2-U1)/R12-(U2-U3)/R23;
                            U3 - U2   U2   U1 - U2
(%o3)                   0 = ------- - -- + ------- + I0
                              R23     R2     R12
(%i4) eq3:0=I0-U3/R3-(U3-U1)/R13-(U3-U2)/R23;
                             U3   U2 - U3   U1 - U3
(%o4)                  0 = - -- + ------- + ------- + I0
                             R3     R23       R13
(%i5) a:matrix(algsys([eq1,eq2,eq3],[U1, U2, U3]));
(%o5) matrix([[U1 = (((3 I0 R1 R2 + 2 I0 R1 R12) R23
 + (3 I0 R1 R13 + 3 I0 R1 R12) R2 + I0 R1 R12 R13) R3
 + (2 I0 R1 R13 R2 + I0 R1 R12 R13) R23 + I0 R1 R12 R13 R2)
/(((R2 + R12 + R1) R23 + (R13 + R12) R2 + (R12 + R1) R13 + R1 R12) R3
 + ((R13 + R1) R2 + (R12 + R1) R13 + R1 R12) R23
 + ((R12 + R1) R13 + R1 R12) R2), U2 = 
(((I0 R12 + 3 I0 R1) R2 R23 + ((2 I0 R12 + 3 I0 R1) R13 + 3 I0 R1 R12) R2) R3
 + ((I0 R12 + 2 I0 R1) R13 + I0 R1 R12) R2 R23)
/(((R2 + R12 + R1) R23 + (R13 + R12) R2 + (R12 + R1) R13 + R1 R12) R3
 + ((R13 + R1) R2 + (R12 + R1) R13 + R1 R12) R23
 + ((R12 + R1) R13 + R1 R12) R2), U3 = 
((((I0 R13 + 3 I0 R1) R2 + (I0 R12 + I0 R1) R13 + 2 I0 R1 R12) R23
 + ((2 I0 R12 + 3 I0 R1) R13 + 3 I0 R1 R12) R2) R3)
/(((R2 + R12 + R1) R23 + (R13 + R12) R2 + (R12 + R1) R13 + R1 R12) R3
 + ((R13 + R1) R2 + (R12 + R1) R13 + R1 R12) R23
 + ((R12 + R1) R13 + R1 R12) R2)]])
(%i6) b:matrix(algsys([eq1,eq2,eq3],[R12, R23, R13]));
(%o6)                   [ [R12 = 0, R23 = 0, R13 = 0] ]
(%i7) 
Run Example
algexact:true;
(%o1)                                true
(%i2) algsys([x^3+(cos(alpha)/sin(alpha)/5.49)*x^2-(cos(alpha)^2*241.27/%pi/75000/5.49^2)=0],[x]);
                              2
(%o2) [[x = expt((sqrt(37) cos (alpha)
                     3
      55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
 sqrt(-----------------------------------------------------))
                           sin(alpha)
             3/2
/(56159001 61    %pi sin(alpha))
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)  1
 - ------------------------------------------------------------------, -)
                                            3                          3
                    1032509122684461 %pi sin (alpha)
             2                     2                          2
 + (10000 cos (alpha))/(2712609 sin (alpha) expt((sqrt(37) cos (alpha)
                     3
      55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
 sqrt(-----------------------------------------------------))
                           sin(alpha)
             3/2
/(56159001 61    %pi sin(alpha))
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)
 - ------------------------------------------------------------------, 
                                            3
                    1032509122684461 %pi sin (alpha)
1     100 cos(alpha)
-)) - ---------------], 
3     1647 sin(alpha)
      sqrt(3) %i   1                    2
[x = (---------- - -) expt((sqrt(37) cos (alpha)
          2        2
                     3
      55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
 sqrt(-----------------------------------------------------))
                           sin(alpha)
             3/2
/(56159001 61    %pi sin(alpha))
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)  1
 - ------------------------------------------------------------------, -)
                                            3                          3
                    1032509122684461 %pi sin (alpha)
             sqrt(3) %i   1     2
 + (10000 (- ---------- - -) cos (alpha))
                 2        2
                                                                    3
                                       2             55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
                           sqrt(37) cos (alpha) sqrt(-----------------------------------------------------)
             2                                                            sin(alpha)
/(2712609 sin (alpha) expt(--------------------------------------------------------------------------------
                                                               3/2
                                                    56159001 61    %pi sin(alpha)
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)
 - ------------------------------------------------------------------, 
                                            3
                    1032509122684461 %pi sin (alpha)
1     100 cos(alpha)
-)) - ---------------], 
3     1647 sin(alpha)
        sqrt(3) %i   1                    2
[x = (- ---------- - -) expt((sqrt(37) cos (alpha)
            2        2
                     3
      55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
 sqrt(-----------------------------------------------------))
                           sin(alpha)
             3/2
/(56159001 61    %pi sin(alpha))
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)  1
 - ------------------------------------------------------------------, -)
                                            3                          3
                    1032509122684461 %pi sin (alpha)
           sqrt(3) %i   1     2
 + (10000 (---------- - -) cos (alpha))
               2        2
                                                                    3
                                       2             55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
                           sqrt(37) cos (alpha) sqrt(-----------------------------------------------------)
             2                                                            sin(alpha)
/(2712609 sin (alpha) expt(--------------------------------------------------------------------------------
                                                               3/2
                                                    56159001 61    %pi sin(alpha)
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)
 - ------------------------------------------------------------------, 
                                            3
                    1032509122684461 %pi sin (alpha)
1     100 cos(alpha)
-)) - ---------------]]
3     1647 sin(alpha)
(%i3) algexact:false;
(%o3)                                false
(%i4) algsys([x^3+(cos(alpha)/sin(alpha)/5.49)*x^2-(cos(alpha)^2*241.27/%pi/75000/5.49^2)=0],[x]);
                              2
(%o4) [[x = expt((sqrt(37) cos (alpha)
                     3
      55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
 sqrt(-----------------------------------------------------))
                           sin(alpha)
             3/2
/(56159001 61    %pi sin(alpha))
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)  1
 - ------------------------------------------------------------------, -)
                                            3                          3
                    1032509122684461 %pi sin (alpha)
             2                     2                          2
 + (10000 cos (alpha))/(2712609 sin (alpha) expt((sqrt(37) cos (alpha)
                     3
      55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
 sqrt(-----------------------------------------------------))
                           sin(alpha)
             3/2
/(56159001 61    %pi sin(alpha))
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)
 - ------------------------------------------------------------------, 
                                            3
                    1032509122684461 %pi sin (alpha)
1     100 cos(alpha)
-)) - ---------------], 
3     1647 sin(alpha)
      sqrt(3) %i   1                    2
[x = (---------- - -) expt((sqrt(37) cos (alpha)
          2        2
                     3
      55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
 sqrt(-----------------------------------------------------))
                           sin(alpha)
             3/2
/(56159001 61    %pi sin(alpha))
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)  1
 - ------------------------------------------------------------------, -)
                                            3                          3
                    1032509122684461 %pi sin (alpha)
             sqrt(3) %i   1     2
 + (10000 (- ---------- - -) cos (alpha))
                 2        2
                                                                    3
                                       2             55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
                           sqrt(37) cos (alpha) sqrt(-----------------------------------------------------)
             2                                                            sin(alpha)
/(2712609 sin (alpha) expt(--------------------------------------------------------------------------------
                                                               3/2
                                                    56159001 61    %pi sin(alpha)
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)
 - ------------------------------------------------------------------, 
                                            3
                    1032509122684461 %pi sin (alpha)
1     100 cos(alpha)
-)) - ---------------], 
3     1647 sin(alpha)
        sqrt(3) %i   1                    2
[x = (- ---------- - -) expt((sqrt(37) cos (alpha)
            2        2
                     3
      55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
 sqrt(-----------------------------------------------------))
                           sin(alpha)
             3/2
/(56159001 61    %pi sin(alpha))
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)  1
 - ------------------------------------------------------------------, -)
                                            3                          3
                    1032509122684461 %pi sin (alpha)
           sqrt(3) %i   1     2
 + (10000 (---------- - -) cos (alpha))
               2        2
                                                                    3
                                       2             55101226617 sin (alpha) - 462214000000 %pi cos(alpha)
                           sqrt(37) cos (alpha) sqrt(-----------------------------------------------------)
             2                                                            sin(alpha)
/(2712609 sin (alpha) expt(--------------------------------------------------------------------------------
                                                               3/2
                                                    56159001 61    %pi sin(alpha)
                       3                         2           3
   231107000000 %pi cos (alpha) - 55101226617 cos (alpha) sin (alpha)
 - ------------------------------------------------------------------, 
                                            3
                    1032509122684461 %pi sin (alpha)
1     100 cos(alpha)
-)) - ---------------]]
3     1647 sin(alpha)
(%i5) 
Run Example
numer:true;
(%o1)                                true
(%i2) algexact:true;
(%o2)                                true
(%i3) g:9.81;
(%o3)                          9.810000000000001
(%i4) x:1.8;
(%o4)                                 1.8
(%i5) y0:2.5;
(%o5)                                 2.5
(%i6) y:0;
(%o6)                                  0
(%i7) eq1: x=v0*t;
(%o7)                             1.8 = t v0
(%i8) eq2: vy=-g*t;
(%o8)                     vy = - 9.810000000000001 t
(%i9) eq3: y=y0-g/2*t^2;
                                               2
(%o9)                         0 = 2.5 - 4.905 t
(%i10) solv1: solve([eq1,eq2,eq3],[t,v0,vy]);

rat: replaced 1.8 by 9/5 = 1.8

rat: replaced 9.810000000000001 by 981/100 = 9.810000000000001

rat: replaced -2.5 by -5/2 = -2.5

rat: replaced 4.905 by 981/200 = 4.905
(%o10) [[t = 0.71392156146353, v0 = 2.52128538646461, 
vy = - 7.003570517957252], [t = - 0.71392156146353, v0 = - 2.52128538646461, 
vy = 7.003570517957252]]
(%i11) eq4: tan(Al)=(vy/v0);
                                           vy
(%o11)                           tan(Al) = --
                                           v0
(%i12) solve([eq4,solv1],[Al]);

rat: replaced 0.71392156146353 by 3641/5100 = 0.71392156862745

rat: replaced 2.52128538646461 by 9180/3641 = 2.521285361164515

rat: replaced -7.00357051795725 by -27461/3921 = -7.00357051772507

rat: replaced -0.7139215614635 by -3641/5100 = -0.7139215686275

rat: replaced -2.52128538646461 by -9180/3641 = -2.52128536116452

rat: replaced 7.003570517957252 by 27461/3921 = 7.00357051772507
(%o12)                                []
(%i13) AlGrd:Al*180/%pi;
(%o13)                       57.29577951308232 Al
(%i14) 

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