solve used to be one of Maple's strongest commands -- it even subsumed simplify in power.  But, over the years, dsolve slowly took over as the most powerful comand.  At the same time, people started realizing that within the framework of differential equations, the toolbox was actually larger than the one for algebraic equations (and most algebraic tools are still available).  So many tasks that one thinks of doing purely algebraically can also be done using differential equations, with perhaps the most surprising one is to factor multivariate polynomials via partial differential equations.

Here we show how to use dsolve to solve parametric equations, ie equations that depend on a parameter -- the availability of a parameter is crucial for this method.  As a first example, let us look at solving a simple version of a cubic equation: x^3+b*x+a=0.  This actually has 2 parameters, so let us choose a as the one to concentrate on.  We want to express this equation as a differential equation -- and luckily there is a command for just that, namely gfun[algeqtodiffeq].  We can pass the result directly to dsolve.

eq1 := x^3 + a*x + b = 0:
de := gfun[algeqtodiffeq](eq1,x(b)):

and we get the result as

                       2    3
  x(b) = -1/3 RootOf(_Z  + a , index = 1)

                        2      2         1/2
        (3 RootOf(_Z (_Z  + a))  + 2 a) 3

                            3 3    b          /  2
        sin(1/3 arctan(------------------))  /  a
                            3       2 1/2   /
                       (-4 a  - 27 b )

                        2                           3 3    b
         + RootOf(_Z (_Z  + a)) cos(1/3 arctan(------------------))
                                                    3       2 1/2
                                               (-4 a  - 27 b )

The first thing to notice is that this is not Cardano's formula, but the much-better trigonometric version of the cubic formula! The second thing we notice is that we can ``see'' the 3 different branches , and the choice that depends on the initial value of a.

Why show this technique now, other than it is fun?  Because of an older post on solve, asking to solve a exp+trig problem.  We can do the same thing here, but we have to use PDEtools[dpolyform] instead.  The command sequence is then

eq3 := exp(-1/k*t)*k/(1+k^2)+(-cos(t)*k+sin(t))/(1+k^2):
ee := eval(eq3,t=t(k)):
de := PDEtools[dpolyform](ee=0,t(k),no_Fn):

which does give a closed-form, with 2 arbitrary constants (as the DE was of order 2). We need initial conditions to get rid of those constants. We can work out easily enough that t(0)=0, and with a little more work that D(t)(0) = RootOf(1-exp(_Z)+_Z*exp(_Z)), which is in fact LambertW(_NN1, -exp(-1))+1. Passing that, as is, as initial conditions to dsolve, I gave up after a few minutes. But with some perseverance, it is possible that one might be able to find a closed-form for this equation!


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