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What does it mean when solving equation and only brackets come back in maxima

What does it mean when solving equation and only brackets come back in maxima?

I tried to solve for phi and it keeps phi on both sides. I try to solve for A and it comes back with brackets Any ideas why?

Here's the equation

eq2:acos(A*cos((2*%pi*f*t+phi)/A) -2*%pi*f*t)=phi;

enter image description here

wxWidgets: 3.0.0 Unicode support: yes Maxima version: 5.32.1

Even when I add values in it still refuses to get phi onto one side

enter image description here

Solution

  • because this is transcendent equation.

    • you can not solve native transcendent equations algebraically
    • if they can be changed by some math trick/substitution to non transcendent form
    • then they are not native transcendent equations.

    This type of equations are solvable numerically

    • so for that you have to know the A or phi value and get the other one
    • there are many numerical solutions to it
    • most cases use some iteration process to improve accuracy
    • but you can not get result as function
    • you find only solution points per specific known input points
    • here an example of one similar equation: solving Kepler`s equation just look at bullet 3 there
    • you can use such iterative process only if your equation meet some conditions
    • like the signum and monotonicity of first/second derivations etc ...

    If you need functional output at some finite interval

    • you can approximate equation by polynomial, Taylor/Furrier series, or whatever else
    • and get the algebraic solution for such form instead