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,
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...
Dear friends,
it my first post.
As I see, this form is helpful for the pleople who needs help on many topics.
Before talking about my problem, I would like to say I can code some programs in Visusal Basic 6.0, Matlab and Maple.
I have coded a program in Maple, but I have just learned...