(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 10.2' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       158,          7]
NotebookDataLength[      5588,        215]
NotebookOptionsPosition[      4947,        187]
NotebookOutlinePosition[      5288,        202]
CellTagsIndexPosition[      5245,        199]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell[BoxData[
 RowBox[{"pde", "=", 
  RowBox[{
   RowBox[{
    RowBox[{"D", "[", 
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"u", ",", "t"}], "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"u", ",", "2"}], "}"}]}], "]"}], "+", 
    RowBox[{"D", "[", 
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"u", ",", "t"}], "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"t", ",", "2"}], "}"}]}], "]"}]}], "\[Equal]", 
   "0"}]}]], "Input",
 CellChangeTimes->{{3.689855177611949*^9, 3.6898552194356227`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{
    SuperscriptBox["f", 
     TagBox[
      RowBox[{"(", 
       RowBox[{"0", ",", "2"}], ")"}],
      Derivative],
     MultilineFunction->None], "[", 
    RowBox[{"u", ",", "t"}], "]"}], "+", 
   RowBox[{
    SuperscriptBox["f", 
     TagBox[
      RowBox[{"(", 
       RowBox[{"2", ",", "0"}], ")"}],
      Derivative],
     MultilineFunction->None], "[", 
    RowBox[{"u", ",", "t"}], "]"}]}], "\[Equal]", "0"}]], "Output",
 CellChangeTimes->{3.6898552313072433`*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"pde", "//", "TraditionalForm"}]], "Input",
 CellChangeTimes->{{3.6898552846125374`*^9, 3.6898552857201395`*^9}}],

Cell[BoxData[
 FormBox[
  RowBox[{
   RowBox[{
    RowBox[{
     SuperscriptBox["f", 
      TagBox[
       RowBox[{"(", 
        RowBox[{"0", ",", "2"}], ")"}],
       Derivative],
      MultilineFunction->None], "(", 
     RowBox[{"u", ",", "t"}], ")"}], "+", 
    RowBox[{
     SuperscriptBox["f", 
      TagBox[
       RowBox[{"(", 
        RowBox[{"2", ",", "0"}], ")"}],
       Derivative],
      MultilineFunction->None], "(", 
     RowBox[{"u", ",", "t"}], ")"}]}], "\[LongEqual]", "0"}], 
  TraditionalForm]], "Output",
 CellChangeTimes->{3.6898552933329525`*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"DSolve", "[", 
  RowBox[{"pde", " ", ",", 
   RowBox[{"f", "[", 
    RowBox[{"u", ",", "t"}], "]"}], ",", 
   RowBox[{"{", 
    RowBox[{"u", ",", "t"}], "}"}]}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"{", 
   RowBox[{
    RowBox[{"f", "[", 
     RowBox[{"u", ",", "t"}], "]"}], "\[Rule]", 
    RowBox[{
     RowBox[{
      RowBox[{"C", "[", "1", "]"}], "[", 
      RowBox[{"t", "+", 
       RowBox[{"\[ImaginaryI]", " ", "u"}]}], "]"}], "+", 
     RowBox[{
      RowBox[{"C", "[", "2", "]"}], "[", 
      RowBox[{"t", "-", 
       RowBox[{"\[ImaginaryI]", " ", "u"}]}], "]"}]}]}], "}"}], 
  "}"}]], "Output",
 CellChangeTimes->{3.6898554990505137`*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"DSolve", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{"pde", ",", " ", 
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"u", ",", "0"}], "]"}], "\[Equal]", "0"}], ",", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{
        RowBox[{"D", "[", 
         RowBox[{
          RowBox[{"f", "[", 
           RowBox[{"u", ",", "t"}], "]"}], ",", "t"}], "]"}], "/.", 
        RowBox[{"t", "\[Rule]", "0"}]}], ")"}], "\[Equal]", "2"}]}], "}"}], 
   ",", "f", ",", 
   RowBox[{"{", 
    RowBox[{"u", ",", "t"}], "}"}]}], "]"}]], "Input",
 CellChangeTimes->{{3.689855655206788*^9, 3.6898556941288567`*^9}, {
  3.6898557522077584`*^9, 3.689855846322724*^9}}],

Cell[BoxData[
 RowBox[{"DSolve", "[", 
  RowBox[{
   RowBox[{"{", 
    RowBox[{
     RowBox[{
      RowBox[{
       RowBox[{
        SuperscriptBox["f", 
         TagBox[
          RowBox[{"(", 
           RowBox[{"0", ",", "2"}], ")"}],
          Derivative],
         MultilineFunction->None], "[", 
        RowBox[{"u", ",", "t"}], "]"}], "+", 
       RowBox[{
        SuperscriptBox["f", 
         TagBox[
          RowBox[{"(", 
           RowBox[{"2", ",", "0"}], ")"}],
          Derivative],
         MultilineFunction->None], "[", 
        RowBox[{"u", ",", "t"}], "]"}]}], "\[Equal]", "0"}], ",", 
     RowBox[{
      RowBox[{"f", "[", 
       RowBox[{"u", ",", "0"}], "]"}], "\[Equal]", "0"}], ",", 
     RowBox[{
      RowBox[{
       SuperscriptBox["f", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"0", ",", "1"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"u", ",", "0"}], "]"}], "\[Equal]", "2"}]}], "}"}], ",", "f", 
   ",", 
   RowBox[{"{", 
    RowBox[{"u", ",", "t"}], "}"}]}], "]"}]], "Output",
 CellChangeTimes->{3.689855886305594*^9}]
}, Open  ]]
},
WindowSize->{759, 586},
WindowMargins->{{17, Automatic}, {Automatic, 32}},
FrontEndVersion->"10.2 for Microsoft Windows (64-bit) (July 7, 2015)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[580, 22, 519, 17, 48, "Input"],
Cell[1102, 41, 518, 19, 31, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1657, 65, 136, 2, 48, "Input"],
Cell[1796, 69, 571, 21, 46, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2404, 95, 202, 6, 48, "Input"],
Cell[2609, 103, 491, 16, 31, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[3137, 124, 679, 20, 48, "Input"],
Cell[3819, 146, 1112, 38, 33, "Output"]
}, Open  ]]
}
]
*)

(* End of internal cache information *)