# Determine the function so that the integral holds

• gop
In summary, the student attempted to integrate by parts to change the integral into to the form of a differential equation, but always ended up with a 1/x term in from of the equation. They were able to solve the problem by taking the derivative of the function and plugging it into the integral.
gop

## Homework Statement

Determine all functions $$H:(0,\infty)\to\mathbb{R}$$
so that $$H(x)=\frac{1}{x}\int_{2}^{x}t(3-2H'(t))\ dt,\ \ x>0$$

## The Attempt at a Solution

I tried to integrate by parts to change the integral into to the form of a differential equation, or at least an equation that could be differentiated to yield a differential equation. However; I always end up with a 1/x term in from of my integral thus making it impossible to get only H'(x) and H(x). Basically, I can't eliminate the integral of H(x) from my equation

try to differentiate both parts with respect to x, and see if you can get anything. or just try to integrate the right-handed side and see if you can come up with sth.

I had to multiply the equation with x then the differentiation works out fine
thx

Yeah, after that you should have gotten a differential equation, i think it should be a linear one, which is solved using an integrating factor. So all solutions of that diff. eq are indeed all the functions H(t)=(3/4) x+Cx^-3, if i can remember it well, because i did it yesterday or sth after you posted, and i am not sure if this is the exact answer i got.
But as a check to your answer take the derivative of this function, and plug it in the integral, integrate it and see if you get the same function H(t).

yeah its a linear one and the result I got is H=(3/4)x + Cx^(-1/3)

gop said:
yeah its a linear one and the result I got is H=(3/4)x + Cx^(-1/3)

$$H(x)=\frac{3}{4}x+Cx^{-3}$$ and not

$$H(x)=\frac{3}{4}x+Cx^{\frac{-1}{3}$$

as a means of checking your answer, follow the instructions i gave u in post # 4

These kind of problems are quite interesting, this one too! ...lol...

Last edited:
I tried in maple
$$dsolve(H(x)+x*(diff(H(x), x)) = x*(3-2*(diff(H(x), x))));$$

and got
$$H(x) = \frac{3}{4}x + \frac{_C1}{x^{1/3}}$$

so I guess the 1/3 is correct.
I did the exercise in class today the one thing I forgot was that because I differentiate I relax the constraints and have to require
H(2) = 0

## 1. What is the purpose of determining the function to make the integral hold?

The purpose of determining the function is to find the exact mathematical relationship between the independent and dependent variables that results in the given integral being satisfied. This allows for a more precise and accurate representation of the system or phenomenon being studied.

## 2. What is an integral and how does it relate to determining a function?

An integral is a mathematical concept that represents the area under a curve on a graph. It relates to determining a function because it is used to find the function that satisfies the given integral. By finding the function, we can then use it to calculate the area under the curve.

## 3. What are the steps involved in determining the function for a given integral?

The first step is to identify the limits of integration and the variable of integration in the given integral. Then, use algebraic manipulation and mathematical techniques such as integration by parts or substitution to solve for the function. Finally, check the solution by taking the derivative and plugging it back into the original integral.

## 4. Can multiple functions satisfy the same integral?

Yes, it is possible for multiple functions to satisfy the same integral. This is because there are different mathematical techniques and methods that can be used to solve for the function, and each may result in a different but valid solution. However, it is important to choose the most appropriate function that accurately represents the system or phenomenon being studied.

## 5. Are there any real-life applications for determining a function to make the integral hold?

Yes, there are many real-life applications for determining a function to make the integral hold. For example, in physics, this concept is used to calculate the work done by a varying force, or the amount of fluid flowing through a pipe. In engineering, it is used for finding the center of mass of an object or calculating the amount of material needed for a construction project.

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