Tagged Questions
1
vote
1answer
62 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
33 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
46 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
31 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 ...
1
vote
1answer
277 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
150 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
84 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
207 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
297 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
317 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: ...
