How to obtain an answer of a partial differential equation from a symbolic computation using DSolve

Questions
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1

Given a mathematical expression EQ. I want to obtain either f[x,t] , Xi[1][x,t] or Xi[2][t] using the command DSolve. See below my attempts,
Input: EQ=r Xi[1][x, t] + r x Derivative[1][Xi[2]][t] - \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[x, t] + x^2 \[Sigma]^2 \!\(\*SuperscriptBox[\(f\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, t] - r x \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, t] - 1/2 x^2 \[Sigma]^2 \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"2", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, t]
Input: DSolve[EQ] == 0, f[x, t], {x, t}]

First attempt output: {{f[x, t] -> C[1][t] + Inactive[Integrate][(-2 r Xi[1][K[1], t] - 2 r K[1] Derivative[1][Xi[2]][t] + 2 \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[K[1], t] + 2 r K[1] \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[K[1], t] + \[Sigma]^2 K[1]^2 \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"2", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[K[1], t])/( 2 \[Sigma]^2 K[1]^2), {K[1], 1, x}]}}

Second attempt input: DSolve[EDsI[[5]] == 0, Xi[1][x, t], {x, t}]
Second attempt output: DSolve[r Xi[1][x, t] + r x Derivative[1][Xi[2]][t] - \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[x, t] + x^2 \[Sigma]^2 \!\(\*SuperscriptBox[\(f\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, t] - r x \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, t] - 1/2 x^2 \[Sigma]^2 \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"2", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, t] == 0, Xi[1][x, t], {x, t}]

Third attempt input: DSolve[EDsI[[5]] == 0, Xi[2][t], {t}]
Third attempt output: {{Xi[2][t] -> C[1] + Inactive[Integrate][(-2 r Xi[1][x, K[1]] + 2 \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"0", ",", "1"}], ")"}], Derivative], MultilineFunction->None]\)[x, K[1]] - 2 x^2 \[Sigma]^2 \!\(\*SuperscriptBox[\(f\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, K[1]] + 2 r x \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"1", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, K[1]] + x^2 \[Sigma]^2 \!\(\*SuperscriptBox[\(Xi[1]\), TagBox[ RowBox[{"(", RowBox[{"2", ",", "0"}], ")"}], Derivative], MultilineFunction->None]\)[x, K[1]])/(2 r x), {K[1], 1, t}]}}
Please advise further on what to do if there is no more obtain the correct solution using DSolve fro the expression.

2

The solution with respect to f is perfectly fine. As for the rest, you don’t explain what EDsI is.

3

Pardon me please for my mistake. That EDsI[[5]] must be replaced by EQ.
AS for the solution for f[x,t]], as it appears it is tricky to substitute for further computation. Does that mean it must be integrated first where there is integral, therefore substitute it for further computation?
Please advise.

4

The solution for f[x,t] contains a symbolic integral because, basically, the equation is of the form y'[x] == h[x], whose solutions are the primitives of h[x]. If you know nothing about h[x] you must leave the symbolic integration as it is. If you know Xi[1], for example, you can replace it into the solution formula this way:

DSolve[EQ == 0, f[x, t], {x, t}] /.
  {Xi[1] -> Function[{x, t}, x + t]} //
 Activate

As for the failed solutions with respect to Xi[1] and Xi[2], Mathematica gives back the input because it cannot find a symbolic solution, I suppose.

5

We advise adding notebook of your code or attaching it to avoid any misunderstanding. The code currently is not properly formatted.

6
Open in Wolfram Cloud ↗

I have added a notebook for clarity towards my earlier question. Please may you advise further.

7

Can you clarify what the question was?

8

I substituted f[x,t] in my further computations, everything worked well and obtained Xi[1][x,t] as shown on the notebook.
Nonetheless, for further computation whereby I substituted both f[x,t] and Xi[1][x,t] , I still fail to obtain Xi[2][t] . Similarly, substituting both f[x,t] and Xi[2]t] , I still fail to obtain Xi[1][x, t] . Please advise further.

Practice0.nb (23 KB)

9

I have been debugging to my ability. The last updated line , that is substituting the Xi[1][x,t] into the equation with partial derivatives still not give an answer. Please advise further.

Practice0.nb (32 KB)

10

Try this:

DSolve[EDsI[[6]] == 0, Xi[1], {x, t}]

instead of

DSolve[EDsI[[6]] == 0, Xi[1][x, t], {x, t}]

otherwise you cannot replace into derivatives of Xi[1].