I have deduced a solution eq16 from the PDE problem eq13, eq14 and eq15. Now I want to examine if eq16 is the right solution. The examinations eq17, eq18 and eq19 show that eq16 can't satisfy eq13 and eq14. But I think eq16 should be right one. Please help me with checking the examination. Thanks.
eq12 = (cinf \[Theta]c \[Lambda])/(s + \[Theta]c) - (
E^((lh - x) /Sqrt[Dc ] Sqrt[s + \[Theta]c]) F1 Sqrt[
Dc ] \[Lambda])/
Sqrt[ (s + \[Theta]c)] /. {Sqrt[Dc] F1 \[Lambda] -> A1,
cinf \[Theta]c \[Lambda] -> A2}
(*equation.*)
eq13 = s *q[x, s] - Dp*D[q[x, s], {x, 2}] - pinf + eq12 == 0
(*left boundary condition.*)
eq14 = \[Alpha]1*(q[x, s] /. x -> lh) - (D[q[x, s], x] /. x -> lh) == 0
(*right boundary condition.*)
eq15 = (D[q[x, s], x] /. x -> +\[Infinity]) == 0
(*a solution to be examined.*)
eq16 = (-A2 + pinf (s + \[Theta]c))/(s (s + \[Theta]c)) - (
A1 Dc E^((lh - x) Sqrt[(s + \[Theta]c)/
Dc]) (\[Alpha]1 Sqrt[Dp (s + \[Theta]c)] + Sqrt[
s (s + \[Theta]c)]))/((Sqrt[s] +
Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s +
Dp (s + \[Theta]c))) + (E^((s (lh - x))/Sqrt[
Dp s]) (Sqrt[
Dp] \[Alpha]1 (Dc s - Dp (s + \[Theta]c)) (-A2 +
pinf (s + \[Theta]c)) +
A1 s (Sqrt[Dc Dp] s + Sqrt[Dc Dp] \[Theta]c +
Dc \[Alpha]1 Sqrt[Dp (s + \[Theta]c)])))/(s (Sqrt[s] +
Sqrt[Dp] \[Alpha]1) (s + \[Theta]c) (-Dc s +
Dp (s + \[Theta]c)))
(*examine eq16.*)
q1[x_, s_] := eq16
eq17 = FullSimplify[
eq14 /. {q -> (q1[#1, #2] &)}, (lh - x) < 0 && s > 0 &&
s + \[Theta]c > 0 && Dp > 0 && Dc > 0 && \[Alpha]1 > 0] // Normal
eq18 = FullSimplify[
eq15 /. {q -> (q1[#1, #2] &)}, (lh - x) < 0 && s > 0 &&
s + \[Theta]c > 0 && Dp > 0 && Dc > 0 && \[Alpha]1 > 0] // Normal
eq19 = FullSimplify[
eq15 /. {q -> (q1[#1, #2] &)}