Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integro-differential-equations]

An integro-differential equation is an equation involving both the integrals and derivatives of a function. The solution to an integro-differential equation is a function which satisfies the original equation.

An integro-differential equation is an equation involving both the integrals and derivatives of a function.

Solution methods include differentiating to obtain a purely differential equation and transform methods such as the Laplace Transform.

Integro-differential equations appear in applications such as RLC linear circuits, when finding the current.

145 questions
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How to solve partial integro-differential equation?

Suppose the following partial integro-differential equation for a function $u(x,t)$ with $t\geq0$, $x \in [0,L]$: $\partial_t u = \partial_{xx} u + f(u,\lambda)$ $\lambda = B\left(u_0 - \int_{x=0}^L {u(x,t)dx} \right) = B \left(u_0 - \right)$…
Name
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How to approach the following differential equation

I have a differential equation of the form $$ \frac{\mathrm{d}g}{\mathrm{d}x} = f(x) + \int_0^x g(y)f(x-y)\mathrm{d}y + \alpha g(x). $$ $f$ is a monotonous decreasing function, satisfying $\int_0^\infty f(x)\mathrm{d}x = 1$. To solve it, I thought…
ck1987pd
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Solving $\frac{\partial}{\partial t} f =hf+ h \int \mathrm {d} i\, h f$

I'm looking for the solution of partial differential equation $$\frac{\partial}{\partial t} f(i,t) =\left(a f(i,t) + b\int_0^\infty \mathrm {d} i\, h(i) f(i,t)\right)h(i)$$ Where $$f(i, 0)=1, \int_0^\infty\mathrm {d} i\, h(i)=1$$ Interesting special…
Yaroslav Bulatov
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An integro differential equation involving $f$,$f_h$ and second derivative of $f$.

Let $f_h$ be the Hilbert transform of the real function $f$. I need some help solving this integro differential equation : $$\alpha f_h(x) + \beta f''(x) = f(x)$$ A simple sinusoid doesn't seem to fit the bill. Kind of puzzzling for me to even guess…
Rajesh D
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Help to solve the integro-differential equation $y'(x)=-k\frac{y^2(x)}{x^2}\int_0^xt^2y(t)\,dt$

I have this differential integral equation from a physics problem $y'(x)=-k\frac{y^2(x)}{x^2}\int_0^xt^2y(t)\,dt$ and i dont know how to solve such equation. Edit 3 (the third time's a charm): Comment: I was wrong at the time of calculating $g(r)$…
fabri bazzoni
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differential equation contains definite integral

I am stuck on solving the following differential equation which contains a definite integral that I don't know how to deal with: $$ f^{\prime\prime} + a^2 f - b\int_0^L f(t) \, dt = c$$ The boundary condition is $f(0)=0$ and $f(L)=R$. Anyone help me…
wallen
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How to solve this integro-differential equation?

I came across this integro-differential equation to solve $$\frac{du(x;t)}{dt}=-\lambda\int_0^xu(\xi;t)\;d\xi\tag{1}$$ under the initial condition $u(x;0)=f(x)$ where $x$ is a parameter, $\lambda$ is a constant, and $0
MasterYoda
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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 $\phi'''(t) + \phi''(t) + \phi'(t) = 0$ through differentiation. I noticed that…
user125980
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Integro-differential equation of one function of one variable with periodic condition

I'm doing physics research and I have been able to boil down my problem to finding the general solution for $\xi(u)$ defined from 0 to 1. I am looking for periodic solutions, so $\xi(0)=\xi(1)$. I have no experience with these types of diff eqs so I…
Nick Mazzoni
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Integro-Differential equation from my Complex analysis exam

My recent complex analysis exam had the following problem as the last question, which I had a hard time solving. The problem Use the Laplace transform to solve the following differential equation for…
Lbark05
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A difficult game theory riddle

Suppose a master hires $n$ servants who work for him. At the end of the day, the service come to him and request a wage for their service. They are able to request a wage up to $\$1$, but no more. With $1/2$ probability, the master is in a good mood…
user180743
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When is the solution to this integro-differential equation 0?

Consider the integro-differential equation $$ \frac{df(t)}{dt} = - \int_0^t ds \ g(s) f(t-s) $$ subject to the initial condition $f(0)=0$ and where $g$ is a known function. My Question: Is the solution to the above always $f(t)=0$? The way I've…
QuantumEyedea
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ODE: $y''y+ax+by+c=0,y=k\pm\sqrt2\int\sqrt{a\int\ln(y)dx-(ax+c)\,\ln(y)-by+K}dx,\int\frac{dy}{\sqrt{K-(ax+c)\,\ln(y)+a\int\ln(y)dx-by}}=k\pm\sqrt2x$

Imagine we had a differential equation like: $$y’’-\frac xy=0$$ Now let’s standardize the signs. Note we do not need a constant for the first term because of the zero product property. We can generalize any way we want, but this way of adding…
Тyma Gaidash
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How to solve differential equation involving an integral

The integro-differential equation takes the form $$c′(t)+iac(t)+\int_{0}^{\infty} f(t−\tau)c(\tau)d\tau=0$$ in which $$f(t−\tau)=\frac{\pi}{4}∫_{0}^{\infty} J(\omega)e^{−i\omega(t−\tau)} d\omega,$$ $$J(ω)=2\pi \alpha…
zxman
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Malthus Model - Solution Differential Equation

I have this equation of a time dependent Malthus model with a term representing a time dependent immigration: $$N'(t)=r(t)N(t) + m(t)$$ with $r(t)$ and $m(t)$ both continuous and periodic with Period $T$. I have to prove that the function $$N_\infty…
LaRausi
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