2
votes
2answers
47 views

To solve non-linear Integro-differential equation

I am just begin to study integral equations, in which i come with following problem regarding second kind Volterra non-linear integro-differential equation, $$u'(x)=-1+\int_{0}^{x}u^{2}(t)dt$$ with ...
0
votes
1answer
55 views

Prove existence and uniqueness of differential/integral equation

This is a homework question, so I'm essentially asking for hints and not answers due to academic honesty concerns. The course is in real analysis (baby rudin) and it is essentially chapter 9 which ...
1
vote
0answers
40 views

Definite Integral of Periodic Function Multiplied by another Function

For one part of a problem I am working on, I am trying to show that $y'(t) \geq 1$ for all $t \geq 0$ when $y'= 1- \int^t_0 g(s)y(s) ds$ When $g(t)$ is periodic, $g(t) <0$ for all $t$, and ...
1
vote
2answers
77 views

Differentiating a convolution integral

I'm trying to turn the integro-differential equation $\phi'(t) + \phi(t) = \int_0^t \sin{(t - \xi)} \, \phi(\xi) \, \mathrm{d} {\xi}$ into the differential equation $\phi'''(t) + \phi''(t) + ...
2
votes
1answer
206 views

The integral equation $y(x)=x-\int_1^x xy(t)dt$ [closed]

The integral equation $$ y(x)=x-\int_1^x xy(t)dt \tag{$y\in C^1[1,\infty)$}$$ has the solution $y=x(1-\log x)$ $y=xe^{\left(x-\frac{1}{2}\right)}(x-1)+x$ $y=xe^{1-x^2}+x$ $y=x-x \cdot e^{x^2}+ex$ ...
0
votes
0answers
164 views

Prove uniqueness of solution for the second kind Volterra integral equation

Let $ K \in C^0([a,b] \times [a,b] ,\mathbb{R})$ and $f \in C^0([a,b],\mathbb{R})$. If we consider the Volterra integral equation of second kind $\phi(x) = f(x) + \lambda \int_a^xK(x,t)\phi(t)dt$, ...
0
votes
2answers
340 views

Find the eigenvalues and eigenvectors of an integral operator

I need to find the eigenvalues e eigenvectors of this integral. $$\int_{0}^{1} K(x,y)\phi (y)dy,$$ where $K(x,y)=x(1-y),\; 0 \le x\le y \le 1$ and $K(x,y)=y(1-x),$ $0\le y\le x \le 1$ I ...
1
vote
1answer
369 views

Writing equivalent first order differential equation and initial condition

I have another homework question that I'm struggling a bit to understand exactly what I'm asked to do. I understand what an initial condition is, but I'm not quite sure how I specify such a ...
2
votes
1answer
43 views

Trying to show that equation has a single solution using Banach space Theorems

How do I show that $f(x) = \int_0^1 e^{-sx}\cos(\alpha f(s))~ds, $ $0\leq x\leq1$, $0\le\alpha\le1$ has a single solution. Using Banach space Theorems like Contraction mapping theorem? Thanks for ...
5
votes
1answer
70 views

How to solve following differential equation?

$$ \int \limits_{0}^{\infty}\sqrt{1 + y'^{2}(x)}dx = 2 \sqrt{x} + y \qquad (.1) $$ The solution is $$ 3y = x\sqrt{x} - 3\sqrt{x} . $$ I don't know how to solve this type of equations. Also I don't ...
0
votes
0answers
43 views

Show a solution to $y(x)=g(x)+\int\limits_{0}^{x}k(x,t,y(t))dt$ exists under certain assumptions on $k(x,t,z)$ and $g(x)$.

I got this homework question that I am stuck on. Let $J = [0, a]$ (with $a > 0$ fixed). Let $g(x)$ be a function which is continuous at all $x \in J$ and let $k(x, t, z)$ be a function which is ...
3
votes
2answers
681 views

Volterra integral equation of secong type solve using resolvent kernel

Solve the integral equation $$ y(t)= f(t) + \lambda \int_{0}^{t} (t-s) y(s) ds $$ where $f$ is continuous using the method of finding the resolvent kernel and Newmann series. Here it is what I ...
2
votes
1answer
224 views

Volterra integral equation of second type

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 mthod of ...
1
vote
1answer
121 views

Comparison between solutions of ODE

Could anyone help on the following problem? Let R(t) be the solution to the integral equation: $R(t)=1+\int_{0}^{t}\frac{1}{R(s)}ds$, namely $R(t)=\sqrt{2t+1}$. Assume that X is continuous and ...
4
votes
1answer
279 views

How can I solve this integral equation in terms of Hermite polynomials?

It must be proven that the solution of the integral equation $$f(x)=\int_{-\infty}^{+\infty} e^{-(x-t)^2} g(t)dt$$ is $$g(x)=\frac{1}{\sqrt{}\pi}\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{2^nn!} H_n(x)$$ ...
0
votes
3answers
386 views

How can I solve this integral equation using characteristic values and eigenfunctions?

$$ f(x)= \int_0^1 e^{|x-t|} f(t) \, dt+x-1 $$ I can't solve it, because I can't find the boundary conditions?
7
votes
2answers
446 views

Solve $f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$

I need to solve this: $\ f(x) = \lambda \int\limits_{0}^1(\max(x,t)+xt)f(t)dt$. Rewriting it as: $\ f(x) = \lambda(\int\limits_0^x x(t+1)f(t)dt + \int\limits_x^1 t(x+1) f(t)dt)$. 1st derivative: ...