How do I solve a differential equation when one of the arguments requires that the value of one variable is equal to the value of the other variable, when I try this I get this error:
"DSolve::conarg: The arguments should be ordered consistently." for input
e = {Derivative[2, 0][u][x, y] + Derivative[0, 2][u][x, y] == 4,
u[x, x] == 2 x^2,
Derivative[1, 0][u][x, x] == 2 x}
s = DSolve[e, u, {x, y}]
The solution should be like x^2 + y^2
DSolve is unable to solve this problem without assistance. Begin by solving the PDE only,
s = DSolveValue[D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}] == 4, u[x, y], {x, y}]
/. {C[1] -> c1, C[2] -> c2}
(* 2 x^2 + c1[I x + y] + c2[-I x + y] *)
and then apply the two auxiliary conditions.
b1 = Simplify[(s /. y -> x) == 2 x^2]
(* c1[(1 + I) x] + c2[(1 - I) x] == 0 *)
b2 = Simplify[(D[s, x]) == 2 x] /. y -> x
(* 2 x + I Derivative[1][c1][(1 + I) x] - I Derivative[1][c2][(1 - I) x] == 0 *)
These two expressions then can be solved to obtain
Flatten@Solve[{D[b1, x], b2}, {Derivative[1][c1][(1 + I) x], Derivative[1][c2][(1 - I) x]}]
(* {Derivative[1][c1][(1 + I) x] -> (1 + I) x, Derivative[1][c2][(1 - I) x] -> (1 - I) x}*)
Although these two relations can be integrated with Mathematica to obtain c1 and c2, the solutions are obvious.
{(c1[(1 + I) x] -> ((1 + I) x)^2/2 + c11), (c2[(1 - I) x] -> ((1 - I) x)^2/2 + c22)}
where c11 and c22 are constants. Next, replace (1 + I) x by (y + I x) and (1 - I) x by (y - I x).
ss = {(c1[(1 + I) x] -> ((1 + I) x)^2/2 + c11) /. x -> (y + I x)/(1 + I),
(c2[(1 - I) x] -> ((1 - I) x)^2/2 + c22) /. x -> (y - I x)/(1 - I)}
Finally, c11 and c22 are determined by
b1 /. (ss /. y -> x) /. Equal -> Rule
(* c11 + c22 -> 0 *)
Simplify[s /. ss] /. %
(* x^2 + y^2 *)
A lengthy process to obtain a simple answer.
Alternative Approach
The auxiliary conditions imposed on the PDE are, in effect, initial conditions on the line x - y == 0. Consequently, changing coordinates so that these initial conditions lie on a coordinate line, say a == 0 seemed productive. The transformation can be accomplished simply, using Kuba's function, DChange.
eq = DChange[D[u[x, y], {x, 2}] + D[u[x, y], {y, 2}] == 4,
{a == x - y, b == x + y}, {x, y}, {a, b}, {u[x, y]}] /. u -> v
(* Derivative[0, 2][v][a, b] + Derivative[2, 0][v][a, b] == 2 *)
Since the inverse transform is given by
Flatten@Solve[{a == x - y, b == x + y}, {x, y}]
(* {x -> (a + b)/2, y -> 1/2 (-a + b)} *)
the auxiliary conditions become
eqb1 = (v[a, b] == 2 x^2) /. % /. a -> 0
(* v[0, b] == b^2/2 *)
eqb2 = DChange[Derivative[1, 0][u][x, y] == 2 x, {a == x - y, b == x + y},
{x, y}, {a, b}, {u[x, y]}] /. u -> v /. a -> 0
(* b == Derivative[0, 1][v][0, b] + Derivative[1, 0][v][0, b] *)
given, as desired, at a == 0. Unfortunately,
DSolveValue[{eq, eqb1, eqb2}, v[a, b], {a, b}]
returns unevaluated. Thus, we are again left with solving
s = DSolveValue[{eq}, v[a, b], {a, b}] /. {C[1] -> c1, C[2] -> c2}
(* a^2 + c1[I a + b] + c2[-I a + b] *)
and nothing has been gained from the transformation. For completeness, the solution proceeds as follows.
b1 = Simplify[(s /. a -> 0) == b^2/2];
b2 = Simplify[D[s, a] + D[s, b] == (a + b)] /. a -> 0;
Flatten@Solve[{D[b1, b], b2}, {Derivative[1][c1][b], Derivative[1][c2][b]}]
(* {Derivative[1][c1][b] -> b/2, Derivative[1][c2][b] -> b/2} *)
ss = {(c1[b] -> b^2/4 + c11) /. b -> b + I a, (c2[b] -> b^2/4 + c22) /. b -> b - I a}
(* {c1[I a + b] -> 1/4 (I a + b)^2 + c11, c2[-I a + b] -> 1/4 (-I a + b)^2 + c22} *)
Simplify[b1 /. (ss /. a -> 0)] /. Equal -> Rule;
Simplify[s /. ss] /. %
(* 1/2 (a^2 + b^2) *)
which is equivalent to the result obtained earlier.
Simplify[% /. {a -> x - y, b -> x + y}]
(* x^2 + y^2 *)
