Consider a differential equation of the following type: $$\sum_n a_n \frac{\partial^n T}{\partial x^n}+\sum_m b_m \frac{\partial^m T}{\partial y^m}=0$$ ($a_n$ and $b_m$ are independent of $x$ and $y$). This can be solved by separation of variables and using the fact that a linear combination of known solutions is a new solution. But, what if the differential equation was of the form $$\sum_{m,n}a_{n,m}\frac{\partial^n}{\partial x^n}\left(\frac{\partial^m T}{\partial y^m}\right)=0$$ The differential equation looks similar to the one before it: we can use the principle of superposition here. However, we cannot use the technique of separation of variables to simplify our task. So, how can we solve such a problem?
This question might seem a bit too general, so let me offer a simpler problem: Solve the following differential equation: $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial}{\partial x}\left(\frac{\partial T}{\partial y}\right) + \frac{\partial^2 T}{\partial y^2}=0.$$ (If you are confused about why I am claiming separation of variables does not lead to any progress, take the above equation. If we write $T(x)=X(x)Y(y)$ then we get: $Y\frac{\partial^2 X}{\partial x^2}+\frac{\partial X}{\partial x}\frac{\partial Y}{\partial y}+X\frac{\partial^2 Y}{\partial y^2}=0$. How do we proceed using separation of variables? From here I do not see any obvious way to write it as $F(x)+G(y)=0$)
If $T(x, y)$ is a continuous and smooth function and you have periodic boundary conditions, you can use the Fourier transform to simplify it. Essentially, the method to solve this problem depends on the boundary conditions.
However, this equation can be simplified using the d'Alembert substitution:
$$ u = x - y \tag 1$$ $$ v = x + y \tag 2$$
Now $T$ is a function of $u$ and $v$. By partially differentiating $T(u, v)$, we get:
$$T_x = T_u u_x + T_v v_x \tag 2$$ $$T_y = T_u u_y + T_v v_y \tag 3$$
$$T_{xx} = T_{uu} (u_x)^2 + T_u u_{xx} + T_{vv} (v_x)^2 + T_v v_{xx} \tag 4$$ $$T_{yy} = T_{uu} (u_y)^2 + T_u u_{yy} + T_{vv} (v_y)^2 + T_v v_{yy} \tag 5$$
$$T_{xy} = T_{uu} u_x u_y + T_u u_{xy} + T_{vv} v_x v_y + T_v v_{xy} \tag 6$$
Since $u_{xx} = v_{xx}= u_{yy} = v_{yy}=u_{xy} = v_{xy}=0$ and $u_x = v_x = v_y = 1$, as well as $u_y=-1$, we get:
$$T_{xx} = T_{uu} + T_{vv} \tag 7$$ $$T_{yy} = T_{uu} + T_{vv} \tag 8$$ $$T_{xy} = -T_{uu} + T_{vv} \tag 9$$
When we plug the above expressions into equation $T_{xx} + T_{xy}+T_{yy}=0$ we get the simplified equation:
$$ T_{uu} + 3 T_{vv}=0 \tag{10}$$
We can get rid of the constant $3$ by doing the substitution $v=\sqrt3 w$:
$$ T_{uu} + T_{ww}=\nabla ^2 T = 0 \tag{11}$$
So in the end, the Laplace equation needs to be solved.