Recent content by Aldnoahz

I see that the only reason we cannot substitute in y(c) = 0 as the solution is that y' goes to infinity as y approaches 0? But that only means the tangent at y = 0 tends to a vertical line. If I directly solve it and find the general solution, they still give me unique solutions that don't cross...

If I understand correctly, uniqueness and existence theorem only guarantees solutions in the region where y' and dy'/dy are continuous, so initial conditions outside this region may or may not result in multiple solutions?

Consider y' = 1/sqrt(y) I seem to be able to find a unique solution given the initial condition of the form y(c) = 0, but the theorem says I won't be able to do so, so I am kind of confused. I just want some clarifications. Does the uniqueness and existence theorem say anything about the...

Hi all, I encountered a fairly straightforward problem involving fluid dynamics, but the result I got is very confusing and I seem not to understand where I got wrong. This is the question: A rectangular flat-bottom barge with a bottom area A=50 m2A=50~\text{m}^2A=50 m2 is loaded so that the...

Is it possible to only look at the mass as a system and regard gravity as an external force and have an intuitive sense as how the center of mass would travel?

Hi all, I've been stuck on thinking about a situation where a mass splits in two in midair. The mass originally undergoes a projectile motion with no air resistance, and it splits in two in midair but one of the piece reaches the ground before the other. In this case, I know that the trajectory...

But the two forces don't have to be equal; how can they satisfy the "this results in a pair of equal and opposite forces" criteria to be action-reaction pair?

Thank you for replying. I just want some clarification. If I analyze the horizontal forces on me (assume no friction), are there two forces? One from the compressed spring and the other from the reaction force by me pushing the spring with the force F? If I do work by exerting this force F to...

Hi all, I am quite confused by the forces relating to a horizontlly placed spring when someone directly exerts a force to the compress it. If I exert a force F to the right to compress the spring, which, say, is placed against a wall, the spring exerts an elastic force on me but I don't see how...

Yes I have looked at both pages but both pages only talk about whether mixed second derivatives are equal at a specific point. I want to know whether fxy and fyx will always look the same in their function forms...

Sorry about the confusion. What I want to ask is that, for example, is it possible to have a function f(x,y) such that fxy = x while fyx = y ?

Hi all, I understand that the mixed partial derivative at some point may not be equal if the such mixed partial derivative is not continuous at the point, but are the actual functions of mixed partial derivatives always equal? In other words, if I simply compute the mixed partial derivatives...

Hi FactChekcer, I don't seem to understand the logic leap you made between and What I want to understand is what happens when you set arclength as the parameter and why

Hi all, I have long had this unsolved question about arclength parameterization in my head and I just can't bend my head around it. I seem not to be able to understand why velocity with arclength as the parameter is automatically a unit tangent vector. My professor proved in class that s(s) =...

I am confused about a transition matrix as I need to prove that if matrix A is positive, then A^(m+1) is also positive. However, when calculating the (m+1)th transition, I need to put matrix A on the left side of equation (A^m)x=x to write A(A^m)x=x. This to me represents after m times...