The extremal of the functional $\int_{0}^{\alpha}{\left((y’)^2 – y^2\right)dx}$ that passes through (0,0) and (${\alpha}$,0) has a weak minimum if ${\alpha}$ < $\pi$ strong minimum if ${\alpha}$ < $\pi$ weak minimum if ${\alpha}$ > $\pi$ strong minimum if ${\alpha}$ > $\pi$ Comparing the given functional to the standard form $J = \int_{a}^{b} F(x, y, y^{‘})$, we […]

How to solve this recursive integral? I do not even begin to understand how to solve this recursive integral. This doesn’t even seem possible? $$H{(x, y)} = \int_{t=0}^{t=2\pi} H(\frac{xt}{2},\frac{xt}{2})*2xt dt$$ This is intriguing me. What approaches exist for solving this? EDIT:$$H{(x)} = \int_{t=0}^{t=2\pi} H\big(\frac{xt}{2}\big)*2xt \, dt$$

Integral equation $$y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$$ has: a unique solution for $\lambda \neq \frac{4}{\pi +2}$; a unique solution for $\lambda \neq \frac{4}{\pi -2}$; no solution for $\lambda \neq \frac{4}{\pi +2}$, but the corresponding homogeneous equation has a non-trivial solution; or no solution for $\lambda \neq \frac{4}{\pi -2}$, but the corresponding homogeneous equation […]

Solve the Volterra integral equation of second kind : $$ y(t)= 1 + 2 \int_{0}^{t} \frac{2s+1}{(2t+1)^2} y(s) ds $$ I know two methods for such integral equations: Picard’s method The method of finding the resolvent kernel and the Neumann series I tried using both of these methods but I couldn’t solve it. Which of the […]

Consider a bounded real-valued function $S:\mathbf{R}\to\mathbf{R}$ so that $$\lim_{x\to\infty} \left( S(x) + \int_1^x \frac{S(t)}{t}dt\right)$$ exists and is finite. Can one say that $\lim_{x\to\infty} S(x)=0$?

So the question is to solve the integral equation to find out $y(x)$. $$y(x)+\int_{0}^{x}(x-s)y(s)ds=\dfrac{x^3}{6}$$ So I find out the Resolvent kernel for $(x-s)$ and got $\sin{(x-s)}$, so I conclude that the solution is $$y(x)=\dfrac{1}{6}\int_{0}^{x}s^3\sin{(x-s)}ds$$ But the answer says $$y(x)=\int_{0}^{x}s\sin{(x-s)}ds$$ What did I miss?

Let us assume there exists some infinite order differential equation whose solution is: $$ y= \sum_{n=1}^\infty A_n \exp(p_n^sx) $$ Where $p_n$ is the $n$’th prime. Substituting $ y=\exp(\lambda x)$ as a trail solution and factorizing. The differential equation must be simplified and factorized to: $$ \prod_{j=1}^\infty (\lambda-p_n^s) = 0$$ $$ \implies \prod_{j=1}^\infty (1-\frac{p_n^s}{\lambda}) = 0$$ […]

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such functions $f$. So far, I’ve managed to show that $\int_0^1\!{f(x)^t\, \mathrm{dx}}$ is constant for $t \geq 2$. Help would be appreciated.

Question: let $a\in(0,1)$, and such $f(x)\geq0$, $x\in R$ is continuous on $R$, if $$f(x)-a\int_x^{x+1}f(t)dt,\forall x\in R $$ is constant, show that $f(x)$ is constant; or $$f(x)=Ae^{bx}+B$$ where $A\ge 0,|B|\le A$ and $A,B$ are constant, and the positive number $b$ is such $\dfrac{b}{e^b-1}=a$ My try: let $$f(x)-a\int_x^{x+1}f(t)dt=C$$ then we have $$f'(x)-af(x+1)+af(x)=0,\forall x\in R$$ other idea: let […]

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?

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