# Reduce is not working for my equation

Can anybody help me?

Reduce[{Cos[Sqrt[w]]*Cosh[Sqrt[w]]*L == 1, w > 0}, w]


This procedure takes a long time and is not working. Can anybody help me?

• Reduce[{Cos[Sqrt[w]]*Cosh[Sqrt[w]]*L == 1, w > 0} // TrigToExp, {w}] indicates that Reduce cannot solve the (transformed) equation. It probably won't be able to solve your original one, either. – Michael E2 Jun 30 at 1:34
• Do you have any reason to believe that a closed form solution should exist? Questions like this should always indicate why it is reasonable to expect a solution. – Szabolcs Jun 30 at 10:58

One difficulty seems to have to do with the constant L. If you are willing to assign a value to this, you can solve numerically. For instance:

L = -0.1; NMinimize[(Cos[Sqrt[w]]*Cosh[Sqrt[w]]*L - 1)^2, w]
{0., {w -> 9.01751}}


Using this as a hint, we can try to use the Reduce command with both w and L as variables:

Reduce[{Cos[Sqrt[w]]*Cosh[Sqrt[w]]*L == 1, w > 0, L > 0}, {w, L}]


Now you get a nice answer!

• This gives the same thing as solving for L (Reduce[.., {L}]). The order matters. Using the variables in the order Reduce[..., {L, w}] is closer to OP's code, and it still runs for longer than I waited. – Michael E2 Jun 30 at 1:31

ContourPlot shows the solution {w, L}

ContourPlot[Cos[Sqrt[w]]*Cosh[Sqrt[w]]*L == 1, {w, 0, 10}, {L,-5, 5},
FrameLabel -> {w, L}] 