# Science & engineering math: integro-differential equation

• chatterbug219
In summary, the conversation was about solving a problem involving a convolution of two functions, y'(u) and y(t-u), with a given integral and initial condition. The participants discussed the use of a transform to reduce the problem to a multiplication, specifically the Laplace transform. They also mentioned the need to take the integral of the given functions and the confusion about how to start the problem. The solution involves taking the Laplace transform of both sides and solving for one function.
chatterbug219

## Homework Statement

$\int$ y'(u)y(t-u)du = 24t3
The integral goes from t (top) to 0 (bottom)
With y(0) = 0

## Homework Equations

I want to say it kind of looks like a convolution problem
$\int$ f(u)g(t-u)du
The integral goes from t (top) to 0 (bottom)

## The Attempt at a Solution

I have no idea...

When you see a convolution in a homework problem, you should immediately think of a transform that reduces it to a multiplication. This transform can even handle the derivative easily...

Well if it is convolution then it would just be
F(s)*G(s)
But I was more concerned about whether or not it actually was convolution. Because its y' and y, and those are both completely different, then it would be convolution then?
So I would need to take the Integral of y'(u) and y(t-u) then? And the integral of 24t3?

it is a convolution between to functions, i.e., y'(t) and y(t), their Laplace transform are related, because one is the derivative of the other. So there is only one F(s) to solve for, the other is simply related to this one.

Okay, understandable...but how do I start the problem? I'm still confused about how to start...

chatterbug219 said:
Okay, understandable...but how do I start the problem? I'm still confused about how to start...

What is the Laplace transform of a convolution?

RGV

F(s)G(s)

So "start" by taking the Laplace transform of both sides!

## 1. What are integro-differential equations?

Integro-differential equations are mathematical equations that involve both derivatives and integrals. They are used to model systems in which the behavior of a variable is dependent on both its current value and its past values.

## 2. What is the application of integro-differential equations in science and engineering?

Integro-differential equations have a wide range of applications in various fields of science and engineering, such as fluid dynamics, heat transfer, quantum mechanics, and population dynamics. They are also used in the study of complex systems and in the development of mathematical models.

## 3. How are integro-differential equations solved?

Integro-differential equations can be solved analytically or numerically. Analytical solutions involve finding an exact formula for the solution, while numerical solutions involve approximating the solution using numerical methods such as finite difference, finite element, or spectral methods.

## 4. What are the challenges in solving integro-differential equations?

Integro-differential equations can be challenging to solve due to their complex nature and the presence of both derivatives and integrals. The lack of closed-form solutions for many integro-differential equations also makes it necessary to use numerical methods, which can be computationally intensive.

## 5. How are integro-differential equations related to other types of differential equations?

Integro-differential equations are a generalization of ordinary and partial differential equations. They can be reduced to these simpler types of equations in certain cases, such as when the integral term is equal to zero. Integro-differential equations are also closely related to fractional differential equations, which involve derivatives of non-integer order.

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