# Recent content by matematikawan

1. ### A Boundary Value Problem

You are right Wrobel. I overlooked that U is negative (the assumption). Sorry. The result now is consistent with an example that I have. u" + 4u = x^2 , u(0)=u(1)=0 Solution: u(x) = (2x^2 + sin(1-2x)/sin(1) - 1)/8 Lower bound solution F=4, G=1 U" + 4U = 1 , U(0)=U(1)=0 Solution...
2. ### A Boundary Value Problem

Sorry to come back again to this thread. If I understand correctly, the mapping \mathcal F : W \rightarrow W where W=\{u\in C[0,1]\mid U\le u\le 0\} required the statement if h\le 0 then Ph=x\int_0^1d\xi\int_0^\xi h(s)ds-\int_0^xd\xi\int_0^\xi h(s)ds \le 0 . Also in the equation -U''=F...
3. ### A Boundary Value Problem

Wow! What a solution. Thank you very much Wrobel. I need some time to properly understand the solution. Hope you don't mind if I ask again in case I do have problem understanding the argument. :smile:
4. ### A Boundary Value Problem

Thanks. Interesting suggestion wrobel. So to which DE should I compare my equation ? Although I would prefer f(x) and g(x) to be general, I would still be content if f(x) is a monotonic increasing function since I'll be solving later on a specific DE with f(x) known.
5. ### A Boundary Value Problem

I have a BVP of the form u" + f(x)u = g(x) , u(0)=u(1)= 0 where f(x) and g(x) are positive functions. I suspect that u(x) < 0 in the domain 0 < x < 1. How do I go proving this. I have try proving by contradiction. Assuming first u > 0 but I can't deduce that u" > 0 which contradict that u has...
6. ### Laplace transform to solve a nonhomogeneous equation

Looks like you have a boundary value problem not initial value problem. If your DE is a linear constant coefficient, I think you still can solve it with Laplace transform.
7. ### Runge-Kutta Method - Need help with the calculus

That explain partly. But why can't we also write the first term as f''(f(f)) and the second term as (f',f')f ?
8. ### Runge-Kutta Method - Need help with the calculus

I got the expression from Prof. J.L. Butcher note, an authoritative person in Runge-Kutta method. That term is related to a rooted tree. Just google rooted tree Runge-Kutta for detail.
9. ### Runge-Kutta Method - Need help with the calculus

When deriving the Runge-Kutta Method to solve y'=f(x) we need to use Taylor expansion. Hence we need to differentiate the function many times. y'(x)=f(y(x)) y''(x)=f'(y(x))y'(x) = f'(y(x))f(y(x)) y''' = f''(y(x))(f(y(x)),y'(x)) + f'(y(x))f'(y(x))y'(x) I can understand the second...
10. ### Role of eigenvalues in phase portraits

Euler's formula e^{it}=\cos{t }+ i \sin{t}. So if the eigenvalue is a complex number, the solution will have sinusoidal functions I think.
11. ### Help with solving system of DE's

Assuming that your system can be written as the matrix form \dot{X}=AX+F(t). , e.g. X=[x1 x2]t etc Then the general solution for this equation should be (if I'm not mistaken) X(t)=e^{At}C+e^{At}\int_0^t e^{-As}F(s) ds Your particular solution in this case is then X_p(t)=e^{At}\int_0^t...