That's a strange request given the wide spread of the Cauchy distribution. Why are you using Cauchy(0,1) if you want such a restriction? You could've used Cauchy(0,1/6), for example.

That's quite an interesting discovery. Vote up.

I wonder if the erroneous values are always 0.*I. I confirm that you've thoroughly and meticulously documented a genuine serious bug, likely in CLAPACK sw_dgeevx_ (CLAPACK sw_zgeevx_ seems okay). Until it's known to be fixed, I recommend to always use datatype= complex(sfloat) in place of datatype= sfloat, even if one is not explicitly computing eigenvalues.

@vv Your piecewise trick is brilliant. It's obvious why it works, but I hadn't thought of it. Vote up.

I doubt that it's possible except possibly by very elaborate "surgery" by a human expert in hypergeometric functions.

Three comments though:

1. Richardson's Theorem guarantees that there are, and always will be, expressions (even fairly simple univariate expressions with no singularities) which equal 0 for all complex values of their variables, yet no algorithm can verify that. This is not a joke theorem like Murphy's Law; it's an actual mathematical theorem. See also: zero-equivalence problem. You should always keep this in mind when trying to simplify something to 0.
2. Your command subs(sol, ode) is total nonsense because sol is an implicit solution rather than one solved for y(x).
3. For what it's worth, the hypergeometric solutions can be converted to an integral form with convert(..., Int). This doesn't help much, but it does give you a different perspective.

@vv Thank you (and a vote up). That is more-than-sufficient explanation for me.

@vv To further my mathematical education, would you please elaborate a bit mathematically? Whether or not you elaborate Maple-wise doesn't matter to me. Specifically, how does an integrand that's entire (in the sense of complex analysis) become something with essential singularities? And, also, what is the path of integration in the complex plane? Is a semicircle added to the real interval 0..2*Pi?

You've asked 137 Questions here over the past 6 years. Most of them have been Answered. So why are you still using linalg? Worse yet, why are you mixing LinearAlgebra and linalg? Are you learning anything from our Answers?

Other than that, I don't understand your Question: Do you have an unwanted execution, or an unwanted expression? Please comment on exactly what part is unwanted., because it's not clear to me from reading your worksheet.

@Carl Love I should've included some explanation of my plot command. So, here it is: I created two separate plots and overlaid them with plots:-display. The first of the inner plot commands plots the shaded region; the second plots the curve. Any number of plots in the same dimension (created with any commands, not just plot) can be overlaid with plots:-display.

A single plot command can also handle multiple plots, thus avoiding the need for display. However, if the x-ranges are different, the way that I showed is easiest. But finer control---allowing for both multiple x-ranges and vertical lines within a single plot command---can be achieved by specifying the curves parametrically (i.e., specifying their x- and y-components as separate functions):

plot(
   [
      [x, f, x= 0..1], #shaded (filled) region
      [x, f, x= -1..1], #full curve
      [1, y, y= 0..eval(f, x=1)] #vertical line segment
   ], 
   filled= [true, false, false], thickness= 3,
   labels= [x, ``], labelfont= [TIMES, BOLD, 16],
   legend= [A, f, x=1]
);

If you want to be very fancy, labelling text can be put directly in the plot:

plots:-display(
   plot(
      [
         [x, f, x= 0..1], #shaded (filled) region
         [x, f, x= -1..1], #full curve
         [1, y, y= 0..eval(f, x=1)] #vertical line segment
      ], 
      filled= [true, false, false], thickness= 3,
      labels= [x, ``], labelfont= [TIMES, BOLD, 16]
   ),
   plots:-textplot(
      [
         [0.75, 0.5, typeset(A), font= [HELVETICA, BOLD, 24]],
         [
            0.5, eval(f, x= 0.5), typeset('f'=f),
            #make baseline of text parallel to f's tangent line: 
            rotation= arctan(eval(diff(f,x), x= 0.5)), align= above
         ],
         [1, 1, typeset(x=1), rotation= Pi/2, align= right]
      ],
      font= [TIMES, BOLD, 16]
   )
);

@Oliveira There's a bug in Acer's example, which is totally excusable since he was posting from his phone. The &-operators have a higher precedence (i.e., their position in the order of operations) than all other arithmetic operators. Thus, any arguments containing arithmetic operators need parentheses:

( sin(x)^2 + cos(x)^2 ) &// simplify;

Acer's version cannot handle multi-argument commands. I'll leave it to him to modify that, since his style may avoid parameterless procedures, such as I used.

In what computer/mark-up language is this expression (copied directly from your Question) written?:

∫(-625 R^2 ro (-2 cp3 x1 lambdaopt+sin((pi t)/10) cp2+(16 cp2)/5) (-cos((pi t)/10)+cos(2 pi)+((-t/5+4) x1+(8 t)/25-32/5) pi) lambdaopt pi b11 (e)^(-((16+5 sin((pi t)/10)) cp1)/(10 lambdaopt x1))+625 (-R^2 k11 (cos((pi t)/10))^3+k11 R^2 (cos(2 pi)-((t-20) (x1-8/5) pi)/5) (cos((pi t)/10))^2+((32 sin((pi t)/10) R^2 k11)/5+(281 R^2 k11)/25+4 lambdaopt^2 (a11 x1+a13 x3)) cos((pi t)/10)-(32 k11 R^2 (cos(2 pi)-((t-20) (x1-8/5) pi)/5) sin((pi t)/10))/5+(281 (x1-8/5) (R^2 k11+(100 lambdaopt^2 (a11 x1+a13 x3))/281) pi t)/125) x1^2)ⅆt

@Oliveira There's a long-standing bug in the kernel regarding eval that forces me to make a small change to the operator to handle your example. The new operator is

`&(`:= ()-> args[-1](args[1], args[2..-2]):

Then your eval example would be handled by 

(x^2 - 2*x, x=4) &( eval

@tomleslie The problem does explicitly say that the function should be plotted from -1 to 1, not merely that that be the viewing window.

@Samir Khan Ah, yes, I figured something else was coming. Building the aniticipation, eh?

@dharr But Heaviside(x) <> x throughout any real open interval.

Surely you understand the very limited value of this Post if you don't reveal the process of extracting the equations from from the cursive script, which surely preceded the equations.