rough question in Differential Equation. I'm trying to solve the following system of differential equations, but I couldn't find any method / procedure to obtain the solution. I don't want a comprehensive and complete answer; a hint will suffice.
Notice that $h$ is a fixed real value.
$$\int_{x_i-h}^{x_i+h} dx\int_{y_j-h}^{y_j+h} dy\nabla\phi_{i\pm 1,j}.\nabla\phi_{i,j}=-\frac 13$$
$$\int_{x_i-h}^{x_i+h} dx\int_{y_j-h}^{y_j+h} dy\nabla\phi_{i,j\pm 1}.\nabla\phi_{i,j}=-\frac 13$$
$$\int_{x_i-h}^{x_i+h} dx\int_{y_j-h}^{y_j+h} dy\nabla\phi_{i\pm 1,j\pm 1}.\nabla\phi_{i,j}=-\frac 13$$
$$x_{i\pm 1}=x_i\pm h$$
$$y_{j\pm 1}=y_j\pm h$$
 A: A non-mathematics student asked me to check if the exercise 2.29, page 94 of the book "Classical Electrodynamics, Third Edition, John David Jackson, John Wiley & Sons" is correct or not? In fact the problem is asking to check 5 integrals. The two first are correct but He and also me earned 8/3 instead of -1/3 for the last three ones. At the time he asked his question I saw your question and his are a little more than a few similar! There are two possibilities;
1st, you was asked the same question but you didn't understand it! (Which seems impossible as the question is some easy integrations and no one is asking about differential equations!)
2nd, you are really asking a function which satisfies these three equations. Then if the author of the book hadn't made a mistake at that exercise then the function which is come there is one answer to your question, otherwise (in case answers of those integrations be 8/3) you only need to multiply that function with (-8), then again you have one answer for your question. Therefore in both cases I introduced you one answer.
