Steady State Mixed Boundary Value Problem I have to solve the following one Steady State Mixed Boundary Value Problem and i need some help with the analytical solution! Considering the Partial Differential Equation: $$ {\nabla }^{2}T=100$$ applied in a rectangle $ \Omega$ where $ \Omega :\left\{0<x<1, 0\le y\le 0.5\right\}$ with the boundary conditions which given below: $$ \left.\begin{array}{r}T\left(0,y\right)=40,\\ T\left(x,0.5\right)=40,\\ T\left(1,y\right)=40,\\ \nabla T\left(x,0\right)=500\end{array}\right\} $$
I tried the Seperation of Variables Technique where: $$ T\left(x,y\right)={X}_{\left(x\right)}{Y}_{\left(y\right)}\to \frac{{X}_{\left(x\right)}^{"}}{{X}_{\left(x\right)}}=-\frac{{Y}_{\left(y\right)}^{"}}{{Y}_{\left(y\right)}}=-\lambda ,\lambda \in \mathbb{R} $$ and obtain the following: $$ \left(A\right):\left\{\begin{array}{l}{X}_{\left(x\right)}^{"}=-\lambda {X}_{\left(x\right)}\\ X\left(1\right)=X\left(0\right)=40\end{array},\left(0<x<1\right)\right. $$ and $$ \left(B\right):\left\{\begin{array}{l}{Y}_{\left(y\right)}^{"}=\lambda {Y}_{\left(y\right)}\\ Y\left(0.5\right)=40\end{array},\left(0\le y\le 0.5\right)\right. $$
From the (A): $ {\lambda }_{n}={n}^{2}{\pi }^{2}\space, and \space {X}_{n}\left(x\right)=sin\left(n\pi x\right),n\in \mathbb{N}.$  And from the (B): $ {Y}_{\left(y\right)}^{"}={n}^{2}{\pi }^{2}\cdot {Y}_{\left(y\right)}\space,\space where:{Y}_{\left(y\right)}=C\sinh\left[n\pi \left(0.5-y\right)\right]+Dcosh\left[n\pi \left(0.5-y\right)\right].$
Are that steps correct? How can i go on? Need to decompose the problem first? And what kind of form my General-Analytical Solution will have? Thank you!
 A: It is important to realize that the differential equation you were given is an inhomogeneous one. This means that the right hand side is unequal to zero.
The proper procedure follows two lines. First of all we seek a (simple) solution to the inhomogeneous case. We can easily verify that the following solution works:
$$T(x,y) = Ax^2 + Bx + (50-A)y^2 + Dy + Exy + F$$
Presumably we won't need all $6$ terms, i.e. most of the parameters will turn out to be zero.
Next we seek solutions to the homogeneous case, i.e. with a RHS equal to zero. This problem can be solved by the Separation of Variables Technique. We assume $T(x,y)=X(x)*Y(y)$. This leads to these solutions:
$$sin(Cx)e^{-Cy}, cos(Cx)e^{-Cy}, sin(Cx)e^{Cy}, cos(Cx)e^{Cy}$$
And also the same expressions with $x$ and $y$ reversed. You can now construct the general solution which is a linear combination of all the homogeneous solutions (with different C's) plus the special solution found for the inhomogeneous case.
The final step is to match the solution to the boundary conditions.
