Get the solution of the following differential equation?

Questions
,
1

I am dealing with current problems and solving differential equations.

phi1 [x] was solved using DSolve.

However, I used DSolve to solve phi2 [x] using 4 boundary conditions, but instead of outputting results, It just show my input.

I write the relevant code below(OutPut for phi2 [x] is omitted) and attach the Mathemtica file.

What’s wrong?

ClearAll[phi1, phi2]
eqn1 := D[phi1[x], {x, 2}] == -q nd / esi
bc1 := phi1[tch + tox esi/eox] == vg - vfb
bc2 := phi1[-tox esi/eox] == vg - vfb
DSolve[{eqn1, bc1, bc2}, phi1[x], x]
{{phi1[x] -> (
   eox esi nd q tch tox + esi^2 nd q tox^2 - 2 eox^2 esi vfb + 
    2 eox^2 esi vg + eox^2 nd q tch x - eox^2 nd q x^2)/(
   2 eox^2 esi)}}


eqn2 := D[phi2[x, y], {x, 2}] + D[phi2[x, y], {y, 2}] == 0 
bc3 := phi2[x, 0] == vs - phi1[x]
bc4 := phi2[x, l] == vs + vd - phi1[x]
bc5 := phi2[tch + tox esi/eox, y] == 0
bc6 := phi2[-tox esi/eox, y] == 0 
DSolve[{eqn2, bc3, bc4, bc5, bc6}, phi2[x, y], {x, y}]
2

I’ll give you a very similar case

In[1]:= s = 
 DSolveValue[{v''[x] == -2*a, v[0] == v0, v[x0] == v0}, v, x]


Out[1]= Function[{x}, v0 - a x^2 + a x x0]

In[2]:= bc = {u[x, 0] == u0 + s[x], u[x, y0] == u1 + s[x], 
   u[0, y] == 0, u[x0, y] == 0};
leqn = Laplacian[u[x, y], {x, y}] == 0;

In[4]:= sol = 
 FullSimplify[
  DSolveValue[{leqn, bc}, u[x, y], {x, y}, Assumptions -> x0 > 0]]

Out[4]= Inactive[Sum][(
 4 Csch[(\[Pi] y0 K[1])/x0] Sin[1/2 \[Pi] K[1]]^2 Sin[(\[Pi] x K[1])/
   x0] ((2 a x0^2 + \[Pi]^2 (u1 + v0) K[1]^2) Sinh[(\[Pi] y K[1])/
      x0] + (2 a x0^2 + \[Pi]^2 (u0 + v0) K[1]^2) Sinh[(\[Pi] (-y + 
         y0) K[1])/x0]))/(\[Pi]^3 K[1]^3), {K[1], 1, \[Infinity]}]
3

Thank you. However, Due to my lack of knowledge, I do not know what’s wrong.

4

The function phi1[x] is not defined in the boundary conditions:

bc3 := phi2[x, 0] == vs - phi1[x]
bc4 := phi2[x, l] == vs + vd - phi1[x]
5

DSolve is symbolical solver and can’t find solution to yours PDE. Maybe you can try a NDSolve a numerical solver.
At first you must assign for all unknown parameters a numerical values.

6

Thanks. I’ll try to solve it.

7

Thanks. I’ll try to solve it.