Recent content by bkarpuz

Set \varphi(t):=\int_{0}^{t}\int_{x}^{t}F(x,y)dydx-\int_{0}^{t}\int_{0}^{y}F(x,y)dxdy and show by Leibniz rule (provided that F is continuous) that \varphi^{\prime}(t)\equiv0, i.e., \varphi is a constant function. Then, complete the proof by using \varphi(t)\equiv\varphi(0)=0 for all t.

Dear MHB members, here is my problem. If $\frac{y^{2}-4}{y+3}=\frac{x^{2}-2x-3}{x+2}=\frac{2}{3}$, then $x+y=?$ Thank you. bkarpuz

Dear MHB members, I have the following equation $xy(z_{xx}-z_{yy})+(x^{2}-y^{2})z_{xy}=yz_{x}-xz_{y}-2(x^{2}-y^{2})$. When I transform this into the canonical form via $\xi=2xy$ and $\eta=x^{2}-y^{2}$, I obtain...

Here is the complete proof. Proof. Existence. Pick some $t_{1}\in[t_{0},\infty)$, and consider the differential equation $\begin{cases} x^{\prime}(t)=p(t)x(t)+f(t)\quad\text{almost for all}\ t\in[t_{0},t_{1}]\\ x(t_{0})=x_{0}. \end{cases}$____________________________(1) Now, define the...

Is there a reference rather than Coddington & Levinson - Theory of Ordinary Differential Equations, McGraw Hill, 1955, which presents Carathéodory's existence theorem? Thanks. bkarpuz

Jose27, thank you very much. Here is my approach, please let me know if I am doing wrong. Existence. Pick some $t_{1}\geq t_{0}$, and define the operator $\Gamma:\mathcal{L}^{\infty}([t_{0},t_{1}],\mathbb{R})\to\mathcal{L}^{\infty}([t_{0},t_{1}],\mathbb{R})$ by...

Dear MHB members, Suppose that $p,f$ are locally essentially bounded Lebesgue measurable functions and consider the differential equation $x'(t)=p(t)x(t)+f(t)$ almost for all $t\geq t_{0}$, and $x(t_{0})=x_{0}$. By a solution of this equation, we mean a function $x$, which is absolutely...

Thank you very much PaulRS for your reply. But if you seach in google DLMF: §26.8 Set Partitions: Stirling Numbers and see Equation 26.8 therein, you will find the formula I have mentioned above. Also I have multiplied the table of Stirling numbers with n=k=4, and I got the identity matrix. I...

Dear MHB members, denote by $s_{n,k}$ and $S_{n,k}$ Stirling numbers of the first-kind and of the second-kind, respectively. I need to see the proof of the identity $\sum_{j=k}^{n}S(n,j)s(j,k)=\sum_{j=k}^{n}s(n,j)S(j,k)=\delta _{{n,k}}$. Please let me know if you know a reference in this...