# Tagged Questions

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### How to solve integral equations like this?

sorry for such a non-specific question and lack of research effort, but I'm new to integral equations and don't know where to start. How does one go about solving equations of the form ...
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### Is this Neumann series solution unique?

I have a Fredholm integral equation of the second kind given as $$f(x)=g(x)+\lambda\int_{-\infty}^\infty K(x,y)f(y)dy,$$ where $\lambda\in(0,1)$, the kernel $K(x,y)=\phi(x-y)$ is a Gaussian ...
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### solve a non-linear integral equation by python

I need to solve an integral equation by python 3.2 in win7. I want to find an initial guess solution first and then use "fsolve()" to solve it in python. This is the code: ...
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### An integral equation of the first kind with variable limits

How should one solve the following integral equation to find $F(\tau)$: $$\int_{\tau -T}^{\tau}F(t)X(\tau -t)dt=e^{-(\frac{\tau}{T})^2}$$
I has worked on this problem for a while and still stucked on it. Hopefully someone give me a hint. Consider the following integral equation, find the function $g(r)$: $$\int_0^\infty {K(s,r)g(r)dr} ... 1answer 43 views ### Combination of Integration and Derivative How to solve this equation:$$ \int_0^{\frac{\pi}{2}}\left(a\sin y-b\left(\frac{dy}{dx}\right)^2\right)~dy=c, $$where a, b, and c are constant. Thank you for your help. 1answer 50 views ### Integral equation question If f(x) and f(t) both have the same domain and range, is there a general way to find \int_{0}^{x^2} f(t) dt = f(x) given t? The actual problem tells that t = 9 and f(x) = 5 e \exp{x cos ... 1answer 60 views ### How to prove \int_a^b f(x)\varphi(x)dx=0\Rightarrow f(x)=0 I am doing some reading on the calculus of variations and one of the first examples uses the following theorem: Let f\in C[a,b]. If \int_a^b f(x)\varphi(x)dx=0 for all \varphi\in C[a,b], then ... 2answers 56 views ### Question about integral is equal to zero Suppose we have the following equation$$ \int_0^\infty {f(x,r)g(x)dx} = 0 \quad {\rm for \, all}\, r\in \mathbb{R} $$where the function g(x) does not depend on r, while f(x,r) is function of ... 1answer 54 views ### Characterization of a particular integrable function Let f be a strictly positive function such that \int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1 (i.e., a probability density function with expectation one). Let also g be a nonnegative ... 1answer 101 views ### Integral Equation without solution? working on a physical problem I arrived at the following equation$$ y(x) + A \int_{0}^{x} e^{\lambda (t-x)} y(t) \mathrm{d}t = 0$$and after some struggling (not that easy to apply the basic Laplace ... 1answer 113 views ### Integral equation u(t)=f(t)+a\int_0^t u(s)ds\quad t\geq 0 Let a\in\mathbb R and f\colon [0,1]\to\mathbb R a continuous function. Solve the integral equation$$u(t)=f(t)+a\int_0^t u(s)\mathrm ds,\quad t\geq 0$$and find an explicit formula for the ... 2answers 79 views ### Integration solving problem A integration is given$$x-x_0 = \pm \int_{0}^{\phi(x)}\frac{d\Phi}{\sqrt\frac{\lambda}{2}(\Phi^2-\frac{m^2}{\lambda})} \tag{1}$$The author said that, equation (2) can be written from equation (1) by ... 2answers 357 views ### Integral equation solution: y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t Integral equation$$y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$$has: a unique solution for \lambda \neq \frac{4}{\pi +2}; a unique solution for \lambda \neq ... 1answer 104 views ### How to solve integral equation x(t)-\int_{0}^{1}[\cos (t) \sec (s) x(s)]ds=\sinh (t), 0\leq t\leq 1. I was thinking about the problem that was as follows: The integral equation x(t)-\displaystyle \int_{0}^{1}[\cos (t) \sec (s) x(s)]ds=\sinh (t), 0\leq t\leq 1, has (a)no solution, (b)a ... 1answer 168 views ### Gelfand-Levitan-Marchenko equation how can one solve the integral$$ f(x,y)+K(x,y)+\int_{0}^{x}K(x,t)f(t,y)dt =0$$(1) so$$ q(x)= 2\frac{d}{dx}K(x,x) $$(2)$$ -y''(x)+q(x)y(x)=0 $$(3)$$ y(0)=0=y(\infty) $$q(x)  here is ... 0answers 100 views ### integral equation solution for two functions  f(x)  and  g(x)  and see if they are related given two functios  f(x)  and  g(x)  related by$$\frac{ \Gamma(s-1/2)}{\Gamma(s) \sqrt{ \pi}}\int_{0}^{\infty}dx \frac{g(x)dx}{(x+y)^{s-1/2}}=\int_{0}^{\infty}dx \frac{f(x)dx}{(x+y)^{s}}$$what ... 2answers 72 views ### homogeneous linear differential equation question I was wondering if there is an analytical solution to the following homogeneous linear differential equation$$\dfrac {dM} {dt}=\dfrac {M} {\alpha \left( t\right) }e^{\beta\left( t\right) t} which ...
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 ...