Recent content by Julio1

Show that the nonlinear system $\dot{X_1}=2\cos X_2, X_1(0)=a$ $\dot{X_2}=3\sin X_1, X_2(0)=b$ has a unique solution for the arbitrary constants $a$ and $b$. how to solve this system? Thanks.

Thanks Hallsoflvy :) My answer is $X(t)=\begin{equation} \begin{pmatrix} X_1\exp(t)\\ X_2\exp(at) \end{pmatrix} \end{equation}$ is correct?

Now fix it. Can you help me now?

Find the general solution of the ODE: $\check{X_1}=X_1$ $\check{X_2}=aX_2$ where $a$ is a constant.

Prove that if $L$ is regular, then $L^R=\{w^R, w\in L\}$ is regular. Hello MHB! I need if you can help me with this problem. Thank you.

Hello HallsofIvy. It is understood that $A_i$ are events and $P$ is a measure of probability, i.e.: $P: \mathcal{A}\to [0,1], A\mapsto P(A).$

Let $G$ be a bipartite graph with bipartition $U$ and $W$ such that $|U|\ge |W|$. Is it true that $\alpha(G) =|U|$?The answer is false, but I don't know how to justify it. I would appreciate any help.

Show that $P(\displaystyle\bigcap_{i=1}^n A_i)=\displaystyle\sum_{i=1}^n P(A_i)-\displaystyle\sum_{i<j} P(A_i\cup A_j)+\displaystyle\sum_{i<j<k} P(A_i\cup A_j\cup A_k)-\cdots - (-1)^n P(A_1\cup A_2\cup ... \cup A_n).$ Hello, the Hint is use induction on $n$.

Let $(X,\tau)$ an topological space. Show that $x_n\to_{n\to \infty} x$ if and only if $d(x_n,x)\to_{n\to \infty} 0.$ Hello, any idea for begin? Thanks.

Hello :). I don't can solve this... Can any help me?

Suppose that $u$ is the solution of the Laplace equation $u_{xx}+u_{yy}=0$ in $\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$ $u(x,y)=x$ for all $(x,y)\in \mathbb{R}^2$ such that $x^2+y^2=1.$ Find the value of $u$ in $(0,0).$ Use the property of median value.

Can help me? :(

Thanks Euge :). But for show that $v\in C^2(\Omega_{(a,b)})$ I don't can show that $v$ has continuous derivate? It is necessary for this case?

Hi Euge :). Yes, $n=\text{lcm}(2,3,5)=30.$

Can someone help me out?