I have the following set of equations:


$\text{$b_1$} \cos (\text{$\beta_1$} u)-\text{$b_1$} \cosh (\text{$\beta_1$} u)-\text{$b2$} \cos (\text{$\beta_1$} \theta y)-\text{$d_2$} \cosh (\text{$\beta_1$} \theta y)\sin (\text{$\beta_1$} u)-\sinh (\text{$\beta_1$} u)=0$


$-\text{$b_1$} \sin (\text{$\beta_1$} u)-\text{$b_1$} \sinh (\text{$\beta_1$} u)+\text{$b_2$} \theta \sin (\text{$\beta $1} \theta y)-\text{$d_2$} \theta \sinh (\text{$\beta_1$} \theta y)+\cos (\text{$\beta_1$} u)-\cosh (\text{$\beta_1$} u)=0$


$-\text{$b_1$} \cos (\text{$\beta_1$} u)-\text{$b_1$} \cosh (\text{$\beta_1$} u)+\alpha ^4 \text{$b_2$} \theta ^2 \cos (\text{$\beta_1$} \theta y)-\alpha ^4 \text{$d_2$} \theta ^2 \cosh (\text{$\beta_1$} \theta y)-\sin (\text{$\beta_1$} u)-\sinh (\text{$\beta_1$} u)=0$

... and I would like to find the constants $b_1$, $b_2$ and $d_2$.

I already know the answers:

$b_1 =\frac{\sin \left(\beta _1 u\right)-\sinh \left(\beta _1 u\right)}{\cosh \left(\beta _1 u\right)-\cos \left(\beta _1 u\right)}$

$b_2 =\frac{2 \cos \left(\beta _1 u\right) \left(\cos \left(\beta _1 u\right) \cosh \left(\beta _1 u\right)-1\right)}{\theta \left(\cosh \left(\beta _1 u\right)-\cos \left(\beta _1 u\right)\right) \left(\cos \left(\beta _1 \theta y\right) \sinh \left(\beta _1 \theta y\right)+\sin \left(\beta _1 \theta y\right) \cosh \left(\beta _1 \theta y\right)\right)}$

$d_2 = -b_2 \frac{\cos \left(\beta _1 \theta y\right)}{\cosh \left(\beta _1 \theta y\right)}$

... but I want to prove it with Mathematica:

eq={b1 Cos[u β1] - b2 Cos[y β1 θ] - 
      b1 Cosh[u β1] - d2 Cosh[y β1 θ] + 
      Sin[u β1] - Sinh[u β1], 
     Cos[u β1] - Cosh[u β1] - b1 Sin[u β1] + 
      b2 θ Sin[y β1 θ] - b1 Sinh[u β1] - 
      d2 θ Sinh[y β1 θ], -b1 Cos[u β1] + 
      b2 α^4 θ^2 Cos[y β1 θ] - 
      b1 Cosh[u β1] - 
      d2 α^4 θ^2 Cosh[y β1 θ] - 
      Sin[u β1] - Sinh[u β1]}

I want to find the constants {b1, b2, d2}

Solve[# == 0 & /@ eq, {b1, b2, d2}] // Simplify

... The results are large terms and I have trouble simplifying it. Any help would be highly appreciated !


closed as unclear what you're asking by mikado, MarcoB, Johu, José Antonio Díaz Navas, LCarvalho Oct 26 '18 at 0:40

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  • 2
    $\begingroup$ I would verify that the equations and solutions are consistent by substituting numerical values for all variables. $\endgroup$ – mikado Oct 5 '18 at 22:14
  • 2
    $\begingroup$ Since you believe you know the value of b1,b2,d2 then FullSimplify[expr1==0,{b1=..,b2=.., d2=..}] should have pretty good chances of returning True, but it returns b2 Cos[y β1 θ] + (1 + d2 Cosh[y β1 θ]) Sin[u β1] == 0 which tends to make me think there might be a typo or another error. If I try to do the substitutions for b1,b2,d2 the result is even worse. Are you certain no errors slipped through? I tried to carefully undo all your Latex work to get it back into MMA, but I might have made mistakes doing that too. $\endgroup$ – Bill Oct 6 '18 at 3:55
  • $\begingroup$ Because α does not appear in the solution given, it follows that b2 Cos[y β1 θ] - d2 Cosh[y β1 θ] == 0, which clearly is inconsistent with the solution for d2 in the question. I believe that there are errors in eq. $\endgroup$ – bbgodfrey Oct 6 '18 at 4:18
  • $\begingroup$ @Bill Thank you very much for your comment and your help ! Indeed this might be a mistake on my side. I will check everything again... but this might take a while. $\endgroup$ – james Oct 6 '18 at 7:15
  • $\begingroup$ @bbgodfrey Thanks for your comment. Yes, there is an error somewhere. As I have already told Bill, I will check everything again. However, this might take a while... $\endgroup$ – james Oct 6 '18 at 7:16