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I am trying to solve the following linear system consisting of four unknowns.

a = 1
b = 1.1
c = 1
w = 100
p = 4500
phi[z_, r_] := 
 a5 (8 z^5 - 40 r^2 z^3 + 15 r^4 z) + b5 (2 z^5 - r^2 z^3 - 3 r^4 z) +
   a1 (2 z^4 + z^2 r^2 - 
     r^4) Log[(Sqrt[z^2 + r^2] + z)/(Sqrt[z^2 + r^2] - z)] - 
  a1 6 z/Sqrt[z^2 + r^2] + 
  b1 (r^4 + 2 r^2 z^2 + 
     z^4) Log[(Sqrt[z^2 + r^2] + z)/(Sqrt[z^2 + r^2] - z)]
sr[z_, r_] = 
 Simplify[D[
    v (D[phi[z, r], {r, 2}] + D[phi[z, r], r]/r + 
        D[phi[z, r], {z, 2}]) - D[phi[z, r], {r, 2}], z] - 
   p w^2 r^2/3 - p w^2 (1 + 2 v) (1 + v) z^2/(6 v (1 - v))]
sz[z_, r_] = 
 Simplify[D[(2 - v) (D[phi[z, r], {r, 2}] + D[phi[z, r], r]/r + 
        D[phi[z, r], {z, 2}]) - D[phi[z, r], {z, 2}], z] + 
   p w^2 (1 + 3 v) r^2/(6 v)]
trz[z_, r_] = 
 Simplify[D[(1 - v) (D[phi[z, r], {r, 2}] + D[phi[z, r], r]/r + 
       D[phi[z, r], {z, 2}]) - D[phi[z, r], {z, 2}], r]]
NSolve[Integrate[sr[z, a], {z, -c, c}] == 0 && 
  Integrate[sr[z, b], {z, -c, c}] == 0 && 
  Integrate[trz[c, r] r, {r, a, b}] == 0 && 
  Integrate[sz[c, r] r, {r, a, b}] == 0, {a1, b1, a5, b5}]

I remember last week I managed to get a set of numerical answers, but this week Mathematica reports "Power::infy: Infinite expression 1/0. encountered." and keeps running without giving me a result. I don't think there is any infinity point in the function. Can someone tell me how to revise the code to get the answer? Thank you!

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  • $\begingroup$ I forget to add that I have defined v to be 0.3. $\endgroup$ – frogjerry Jun 26 '19 at 2:35
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I made a slight modification to your code and it gives an answer on 12.0.0 + OSX 10.14.5. You may want to make sure that you have nothing defined for any of the internal variables using ClearAll[]. 

ClearAll[z, v, n, a1, b1, a5, b5];
a = 1;
b = 1.1;
c = 1;
w = 100;
p = 4500;
phi[z_, r_] := 
  a5 (8 z^5 - 40 r^2 z^3 + 15 r^4 z) + 
  b5 (2 z^5 - r^2 z^3 - 3 r^4 z) + 
    a1 (2 z^4 + z^2 r^2 - 
      r^4) Log[(Sqrt[z^2 + r^2] + z)/(Sqrt[z^2 + r^2] - z)] - 
   a1 6 z/Sqrt[z^2 + r^2] + 
   b1 (r^4 + 2 r^2 z^2 + 
      z^4) Log[(Sqrt[z^2 + r^2] + z)/(Sqrt[z^2 + r^2] - z)];
sr[z_, r_] = 
  Simplify[D[
     v (D[phi[z, r], {r, 2}] + D[phi[z, r], r]/r + 
         D[phi[z, r], {z, 2}]) - D[phi[z, r], {r, 2}], z] - 
     p w^2 r^2/3 - p w^2 (1 + 2 v) (1 + v) z^2/(6 v (1 - v))];
sz[z_, r_] = 
  Simplify[D[(2 - v) (D[phi[z, r], {r, 2}] + D[phi[z, r], r]/r + 
         D[phi[z, r], {z, 2}]) - D[phi[z, r], {z, 2}], z] + 
    p w^2 (1 + 3 v) r^2/(6 v)];
trz[z_, r_] = 
  Simplify[D[(1 - v) (D[phi[z, r], {r, 2}] + D[phi[z, r], r]/r + 
        D[phi[z, r], {z, 2}]) - D[phi[z, r], {z, 2}], r]];
NSolve[Integrate[sr[z, a], {z, -c, c}] == 0 && 
  Integrate[sr[z, b], {z, -c, c}] == 0 && 
  Integrate[trz[c, r] r, {r, a, b}] == 0 && 
  Integrate[sz[c, r] r, {r, a, b}] == 0, {a1, b1, a5, b5}]

(* {{a1 -> -((1. (((-267024. + 
           1.95221*10^-10 I) + (21216.8 - 
            9.48377*10^-10 I) v + (384367. + 
            1.45108*10^-9 I) v^2 - (79069.4 + 
            7.10136*10^-10 I) v^3) (((331.813 + 
              8.65498*10^-13 I) - (3.41061*10^-13 + 
               3.38673*10^-13 I) v) (0. + (-10.592 + 
               11.916 v) (5.13*10^7/(-1. + v) + 
               5.*10^6/((-1. + v) v) - (
               2.63*10^7 v)/(-1. + v))) - 
         1. (0. + ((2.61056*10^6 - 1.16415*10^-10 I) + (
               870187. - 2.91038*10^-11 I)/v) (-10.592 + 
               11.916 v)) ((-10.592 + 
               11.916 v) (275.6/(-1. + v) - (
               275.6 v)/(-1. + v)) - 
            52.96 (-(91.12/(-1. + v)) + (149.8 v)/(-1. + v) - (
               58.68 v^2)/(-1. + v)))) - 
      1. (-1. (-1. ((-1.75455 - 
                1.06581*10^-14 I) - (5.07465 + 
                7.88861*10^-31 I) v) (1.37707 - 
               1.67258 v) + ((16.6258 + 
                0. I) - (13.2092 + 
                2.84217*10^-14 I) v) (-10.592 + 
               11.916 v)) ((-10.592 + 
               11.916 v) (275.6/(-1. + v) - (
               275.6 v)/(-1. + v)) - 
            52.96 (-(91.12/(-1. + v)) + (149.8 v)/(-1. + v) - (
               58.68 v^2)/(-1. + v))) + ((331.813 + 
              8.65498*10^-13 I) - (3.41061*10^-13 + 
               3.38673*10^-13 I) v) (-1. (1.37707 - 
               1.67258 v) (-(91.12/(-1. + v)) + (
               149.8 v)/(-1. + v) - (
               58.68 v^2)/(-1. + v)) + (-10.592 + 
               11.916 v) (29.6561/(-1. + v) - (
               173.856 v)/(-1. + v) + (
               144.2 v^2)/(-1. + v)))) (-1. (-52.96 (76. - 
               36. v) - 
            200. (-10.592 + 
               11.916 v)) (0. + ((2.61056*10^6 - 
                1.16415*10^-10 I) + (870187. - 2.91038*10^-11 I)/
               v) (-10.592 + 11.916 v)) + ((331.813 + 
              8.65498*10^-13 I) - (3.41061*10^-13 + 
               3.38673*10^-13 I) v) (0. + (-10.592 + 
               11.916 v) (-3.*10^7 + (
               5.*10^6 (1. + v) (1. + 2. v))/((-1. + v) v))))))/((
    2.15005*10^8 + 8.27372*10^-7 I)/(-1. + 
     v) - ((2.30422*10^9 + 7.2686*10^-6 I) v)/(-1. + 
     v) + ((9.07026*10^9 + 0.0000363434 I) v^2)/(-1. + 
     v) - ((1.5815*10^10 + 0.0000347323 I) v^3)/(-1. + 
     v) + ((1.23808*10^10 - 0.0000679472 I) v^4)/(-1. + 
     v) - ((3.3374*10^9 - 0.000127206 I) v^5)/(-1. + 
     v) - ((2.0946*10^8 + 0.0000546427 I) v^6)/(-1. + 
     v) + ((0. + 2.13791*10^-7 I) v^7)/(-1. + v))), 
b1 -> -(((4.23724*10^6 - 
      3.97001*10^6 I) ((-2.68178 + 
        2.65827 I) + (1.52727*10^16 - 
         1.63007*10^16 I) v - (3.42115*10^16 - 
         3.65144*10^16 I) v^2 - (5.0757*10^16 - 
         5.41736*10^16 I) v^3 + (1.95778*10^17 - 
         2.08956*10^17 I) v^4 - (1.10133*10^17 - 
         1.17546*10^17 I) v^5 - (1.58126*10^17 - 
         1.6877*10^17 I) v^6 + (2.30055*10^17 - 
         2.4554*10^17 I) v^7 - (1.02746*10^17 - 
         1.09662*10^17 I) v^8 + (1.5357*10^16 - 
         1.63907*10^16 I) v^9 - (4.88834*10^14 - 
         5.21739*10^14 I) v^10 + (1. + 0. I) v^11))/((-1. + 
      v) v ((3.37708 - 
        3.27992*10^-14 I) - (0.268331 - 
         1.44042*10^-14 I) v - (4.86114 - 2.53068*10^-14 I) v^2 + 
      1. v^3) ((3.87001 - 
        1.00568*10^15 I) - (33.9987 - 
         1.07779*10^16 I) v + (169.995 - 
         4.24259*10^16 I) v^2 - (162.46 - 
         7.39741*10^16 I) v^3 - (317.821 + 
         5.79108*10^16 I) v^4 + (595.004 + 
         1.56106*10^16 I) v^5 - (255.59 - 
         9.79744*10^14 I) v^6 + (1. + 0. I) v^7))), 
a5 -> -(((1.96393*10^-7 - 
      2.24393*10^-8 I) ((1.7977*10^28 - 

        1.57338*10^29 I) + (2.1689*10^30 - 
         1.89826*10^31 I) v - (1.10834*10^31 - 
         9.70043*10^31 I) v^2 + (2.12511*10^31 - 
         1.85993*10^32 I) v^3 - (1.52482*10^31 - 
         1.33455*10^32 I) v^4 - (1.08663*10^31 - 
         9.51038*10^31 I) v^5 + (3.66366*10^31 - 
         3.20651*10^32 I) v^6 - (4.01472*10^31 - 
         3.51376*10^32 I) v^7 + (2.31305*10^31 - 
         2.02443*10^32 I) v^8 - (6.28849*10^30 - 
         5.50381*10^31 I) v^9 + (3.8131*10^29 - 
         3.33729*10^30 I) v^10 + (4.71739*10^28 - 
         4.12875*10^29 I) v^11 - (9.53322*10^14 - 
         8.7082*10^14 I) v^12 + (1. + 0. I) v^13))/(((331.813 + 
        8.65498*10^-13 I) - (3.41061*10^-13 + 
         3.38673*10^-13 I) v) (-1. + 
      v) v ((3.37708 - 
        3.27992*10^-14 I) - (0.268331 - 
         1.44042*10^-14 I) v - (4.86114 - 2.53068*10^-14 I) v^2 + 
      1. v^3) ((3.87001 - 
        1.00568*10^15 I) - (33.9987 - 
         1.07779*10^16 I) v + (169.995 - 
         4.24259*10^16 I) v^2 - (162.46 - 
         7.39741*10^16 I) v^3 - (317.821 + 
         5.79108*10^16 I) v^4 + (595.004 + 
         1.56106*10^16 I) v^5 - (255.59 - 
         9.79744*10^14 I) v^6 + (1. + 0. I) v^7))), 
b5 -> ((2.58987*10^-6 - 
    2.11676*10^-8 I) ((4.39031*10^26 - 
      5.37159*10^28 I) + (9.38932*10^28 - 
       1.14879*10^31 I) v - (4.26607*10^29 - 
       5.21957*10^31 I) v^2 + (4.98212*10^29 - 
       6.09567*10^31 I) v^3 + (4.92008*10^29 - 
       6.01977*10^31 I) v^4 - (1.55503*10^30 - 
       1.9026*10^32 I) v^5 + (9.13331*10^29 - 
       1.11747*10^32 I) v^6 + (5.72367*10^29 - 
       7.00296*10^31 I) v^7 - (9.48111*10^29 - 
       1.16002*10^32 I) v^8 + (4.28735*10^29 - 
       5.24561*10^31 I) v^9 - (7.30931*10^28 - 
       8.94301*10^30 I) v^10 + (3.86213*10^27 - 
       4.72535*10^29 I) v^11 - (9.72616*10^14 - 
       9.71795*10^14 I) v^12 + (1. + 0. I) v^13))/(((331.813 + 
      8.65498*10^-13 I) - (3.41061*10^-13 + 
       3.38673*10^-13 I) v) (-1. + v) v (-0.888889 + 
    1. v) ((3.37708 - 
      3.27992*10^-14 I) - (0.268331 - 
       1.44042*10^-14 I) v - (4.86114 - 2.53068*10^-14 I) v^2 + 
    1. v^3) ((3.87001 - 
      1.00568*10^15 I) - (33.9987 - 
       1.07779*10^16 I) v + (169.995 - 
       4.24259*10^16 I) v^2 - (162.46 - 
       7.39741*10^16 I) v^3 - (317.821 + 
       5.79108*10^16 I) v^4 + (595.004 + 
       1.56106*10^16 I) v^5 - (255.59 - 
       9.79744*10^14 I) v^6 + (1. + 0. I) v^7))}} *)
|improve this answer
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  • $\begingroup$ I forget to add one line of code where I define v to be 0.3. I have also noticed that while Mathematica is able to give me an answer when v is not defined, it reports "Power::infy: Infinite expression 1/0. encountered" when v is defined. Why is this happening? $\endgroup$ – frogjerry Jun 26 '19 at 2:45
  • $\begingroup$ Did you click on the stack trace? I ask because I get the same divide by zero problem and the place where it is failing is this: [Integral][Integral]Log[( z + Sqrt[1 + z^2])/(-z + Sqrt[1 + z^2])]/((0. + 1. I) - 1. z)^4 [DifferentialD]z [DifferentialD]z The stack trace may be found by clicking on the button with the 3 dots just to the left of the reported error message. And unfortunately, I don't know why it is failing. $\endgroup$ – Mark R Jun 26 '19 at 5:39

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