DSolve with piecewise coefficeints wrong, why?

Question

sorry if this is a duplicate (I am pretty sure) but I was not able to find the question. I am playing around with DSolve with variable coefficients. In the documentation it is said, that DSolve can handle piecewise defined coefficients, see screenshot below.

Now I am comparing DSolve with NDSolve (with the FEM package) and I get the following results

(*Domain*)
xmin = 0;
xmax = 10;
reg = ImplicitRegion[xmin <= x <= xmax, {x}];
(*Field equation*)
feq = D[A[x]*sig[x], x] + n[x];
sig[x_] := Em[x]*eps[x];
eps[x_] := D[u[x], x];
(*Paremeters*)
Em[x_] := 
Piecewise[{{100,xmin <= x < (xmax - xmin)/3}, {1, (xmax - xmin)/3 <= x <= xmax}}];
A[x_] := 7;
(*Inhomogeneity*)
n[x_] := 3*x;
(*Analytical solution*)
conds = {u[xmin] == 0, u[xmax] == 1};
uds = DSolveValue[{feq == 0, conds}, u, {x, xmin, xmax}];
(*Fem solution*)
Needs["NDSolve`FEM`"]
conds = {DirichletCondition[u[x] == 0, x == xmin],DirichletCondition[u[x] == 1, x == xmax]};
ufem = NDSolveValue[{feq == 0, conds}, u, Element[x, reg]];
(*Plot*)
Plot[{uds[x], ufem[x]}, {x, xmin, xmax}, PlotLegends -> "Expressions"]

And the following plot

The results of uds are wrong, while ufem is correct. Already for n=0 I get the plot

where uds is wrong.

Question: do you see where my error in DSolve is? Is DSolve not able to handle piecewise defined coefficients? Thank you!

Answer 1

Maybe I misunderstood something but I don't see any difficulty with DSolve and NDSolve. The results are identical (version 8)

The ODE is solved explicitly

yd[x_] = y[x] /. 
  DSolve[y'[x] + Clip[x]^2 y[x] == 0 && y[0] == 1, y[x], x][[1]]

$\begin{array}{ll} \{ & \begin{array}{ll} e^{-\frac{2}{3}-x} & x\leq -1 \\ e^{-\frac{x^3}{3}} & -1<x\leq 1 \\ e^{\frac{2}{3}-x} & \text{True} \\ \end{array} \\ \end{array}$

NDSolve gives

yn[x_] = y[x] /. 
  NDSolve[y'[x] + Clip[x]^2 y[x] == 0 && y[0] == 1, 
    y[x], {x, -2, 2}][[1]]

$\text{InterpolatingFunction}[\{\{-2.,2.\}\},<>][x]$

Plotting both functions in one plot (with a small difference of 0.1 added to see both curves) shows the identity

Plot[{yd[x], 0.1 + yn[x]}, {x, -2, 2}]
(* 150603_plot _(n)dsolve_clipx.jpg *)

The same goes for other values of y[0], e.g. -1.

Also, the example case of a piecewise forcing function shows agreement between DSolve and NDSolve.

EDIT #1 Your example of a piecewise function

Your equation is solved by DSolve but not by NDSolve

xmin = 0;
xmax = 10;

Em[x_] := 
  Piecewise[{{100, 
     xmin <= x < (xmax - xmin)/3}, {1, (xmax - xmin)/3 <= x <= xmax}}];

ud[x_] = Assuming[0 < x < 10, 
  u[x] /. DSolve[{7 D[u'[x] Em[x], x] + 3 x == 0, u[0] == 0, 
      u[10] == 1}, u[x], x][[1]]]

$\begin{array}{ll} \{ & \begin{array}{ll} \frac{568 x}{105}-\frac{x^3}{1400} & x\leq \frac{10}{3} \\ -\frac{110}{21}+\frac{233 x}{30}-\frac{x^3}{14} & \text{True} \\ \end{array} \\ \end{array}$

Plot[ud[x], {x, 0, 10}]
(* 150604_Plot _ud.jpg *)

This is ok, but NDSolve[] gives the error message with respect to the derivative:

Assuming[0 < x < 10, 
 NDSolve[{7 D[u'[x] Em[x], x] + 3 x == 0, u[0] == 0, u[10] == 1}, 
   u[x], {x, 0, 10}][[1]]]

NDSolve::ndnum: Encountered non-numerical value for a derivative at x == 0.`. >>

$\left\{3 x+7 \left(\left( \begin{array}{ll} \{ & \begin{array}{ll} 0 & x<0\left\|0<x<\frac{10}{3}\right\|\frac{10}{3}<x<10\|x>10 \\ \text{Indeterminate} & \text{True} \\ \end{array} \\ \end{array} \right) u'[x]+\left( \begin{array}{ll} \{ & \begin{array}{ll} 100 & 0\leq x<\frac{10}{3} \\ 1 & \frac{10}{3}\leq x\leq 10 \\ 0 & \text{True} \\ \end{array} \\ \end{array} \right) u''[x]\right)==0,u[0]==0,u[10]==1\right\}$

$Version

(* Out[245]= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" *)