I couldn't find it anywhere, so I decided to write my question here:
I have problems solving this equation:
$$u_{xx} + u_{yy} = 4,$$
subjected to the conditions
$$u(x,x)=2x^2, \quad u_x(x,x)=2x$$
I've solved a similar problem, only there the initial conditions were on $u(0,y)$ and $u_x(0,y)$ and I used Taylor's expansion there:
$$u(x,y)= u(x_0,y) + u_x(x_0, y)(x-x_0)+ \frac{1}{2} u_{xx}(x_0, y)(x-x_0)^2 + ...$$
and there I could plug $x_0 = 0$.
Could you tell me how to use it here?
Given that the initial condition is a quadratic polynomial, and Laplacian is constant, we should expect to find $u$ as a quadratic polynomial. That is, look for $u$ in the form $$u(x,y)=Ax^2+Bxy+Cy^2$$ The PDE gives $A+C=2$, the value of $u(x,x)$ gives $A+B+C=2$, and the value of $u_x(x,x)$ gives $2A+B=2$. Solve for $A,B,C$ and you are done.