不得已，这样提系数有点复杂。实际问题中变量很多。我需要提取所有系数，分别令系数等于0，再得到若干方程。请教各位大神。
代码:
a = v[x, y]
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y] + u[x, y]*
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y] + x*u[x, y];
b = MonomialList[a, {u[x, y],
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y],
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y]}]
b = b /. MapThread[Rule, {{u[x, y],
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y],
\!\(\*SuperscriptBox[\(u\),
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[x, y]}, Table[1, {3}]}]

这样？：
DeleteCases[List @@ Expand@a, u[__]^(_. ) | Derivative[__][u][__]^(_.), {2}]

@xzcyr 的思路就是很直观, 把不感兴趣的变量踢掉
当然还有一个思路, 那就是给变量赋值为1, 那么结果乘下来也就是系数了
其实异曲同工
ReplaceAll[
a // MonomialList, {Derivative[___][u][___] -> 1, u[___] -> 1}]
当然, Derivative[___][u][___] 这个形式是怎么得到的呢?
InputForm看看就行了, 这其实偏微分最坑的地方, 一不小心就会出现bug

不是有CoefficientList嘛
CoefficientList[a, {u[x, y], Derivative[1, 0][u][x, y], Derivative[0, 1][u][x, y]}]
然后可以Flatten一下
Flatten[%]
再把0删掉
DeleteCases[%, 0]
最后列好方程
% == 0 // Thread

这么多人的助力帖子，我当时怎么就忘记感谢和学习了呢，谢谢吧主又给扒出来