I will try to explain this with an analogy. Let's have this equation: x^2 =9 And let's assume I don't know algebraic methods to solve it, so I create a list using excel with different values. And I see that if I put x=4 it doesn't work, if I put x=5 it is even worse and so on. But If I put 3.9 I see it gets closer to 9, if I put 3.8, 3.7 it gets closer and closer so finally I found x=3 just by trial and error. My question is, let's have a differential equation: dy/dx + (dy/dx)^2 = 0 ANd let's say I want to use the previous method to find a dy/dx in a very specific point, let's say the point (0,1). The question is, Can I be sure it will converge? With an algebraic equation even if I don't know a method to get the general solution using radicals I am able to get solutions just by substituting values and watching if they work or not. Can I do the same with differential equations? Just by using different straight lines with different slopes and testing if they work on a very specific point, Can I be sure the values will converge to a solution? Maybe not and I create an immense list of values with excel and I don't see the data geting closer to any slope, i don't know.