I have an electric potential which, through separation of variables, can be written as $$\phi (x,y)= X(x) \cdot Y(y) =\sum_{n=0}^\infty Cn\cdot \cos(k_n x)\cdot \sinh (k_n y)$$ with $C_n $ and $k_n$ constants which come from my boundary conditions. To find the equations for the electric field lines, we can use the fact that: $$\frac{d x}{E_x}=\frac{d y}{E_y}$$ But since the electric potential is written in terms of an infinite sum, the outcome for $y(x)$, after calculating $E_x$ and $E_y$ and eliminating the constants, would be: $$y(x)=\int \frac{Ey}{Ex}dx=\int \frac{\sum_{n=0}^\infty \sin(k_nx)\sinh(k_ny)}{\sum_{n=0}^\infty \cos(k_nx)\cosh(k_ny)}dx$$ Which seems impossible to inegrate. Is there another way to get the equations of the electric field lines in this case?
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Since $\frac{dy}{dx}$ gives us the variation of the curves throughout the plane, we can use it to analysize the electric field lines without finding the full equations.
By evaluating the differential, $\frac{dy}{dx}$, at the system's boundary points, we can then use that information to try to map the electric field lines and verify our results by confronting them with the frontier-conditions.
Eventhough this doesn't allow us to solve the problem analytically, it gives a good insight into how the electric field lines behave.
Alternatively, we can use the information about the electric field itself to map the lines using Python or other programming software.