7.5 Solving Nonalgebraic Equations
The main equations that Solve and related Mathematica TE functions deal with are polynomial equations. In addition to being able to solve purely algebraic equations, Mathematica TE can also solve some equations involving other functions.
After printing a warning, Mathematica TE returns one solution to this equation.
In[1]:= Solve[ Sin[x] == a, x ]
Out[1]= 
It is important to realize that an equation such as 
actually has an infinite number of possible solutions, in this case differing by multiples of 
. However, Solve by default returns just one solution, but prints a message telling you that other solutions may exist.
There is no explicit "closed form" solution for a transcendental equation like this.
In[2]:= Solve[ Cos[x] == x, x ]
Out[2]= 
There seems to be a root near 1.
In[3]:= Plot[{Cos[x], x}, {x, -5, 5}]
Out[3]= 
You can find an approximate numerical solution using FindRoot and giving a starting value for x.
In[4]:= FindRoot[ Cos[x] == x, {x, 1} ]
Out[4]= 
FindRoot will be discussed in more detail in Section 14.1.
Solve can also handle equations involving symbolic functions. In such cases, it again prints a warning, then gives results in terms of formal inverse functions.
Mathematica TE returns a result in terms of the formal inverse function of f.
In[5]:= Solve[ f[x^2] == a, x ]
Out[5]= 
