# Dirac delta function (DE problem) solved

• ESPCerpinTaxt
In summary, the problem is to determine the deflection of a uniform beam with a concentrated load at the middle, using the Laplace transform and given boundary conditions. The solution involves finding the constants c1 and c2, and using the final equations to obtain the deflection function y(x).
ESPCerpinTaxt
NOTE: I actually found the correct answer while I was typing this and since I already had it typed, I figured i would post anyway. mods you can do with it as you please or leave it for reference. thanks

Here's the problem:

A uniform beam of length L carries a concentrated load $$w_{0}$$ at $$x=\frac{1}2{L}$$. The beam is embedded at its left end and is free at its right end. Use the Laplace transform to determine the deflection y(x) from $$EI\frac{d^4y}{dx^4}=w_{0}\delta(x-\frac{L}2)$$ where $$y(0)=0, y'(0)=0, y''(L)=0, y'''(L)=0$$.

Here is what I did: let $$y''(0)=c_{1}, y'''(0)=c_{2}$$

$$EI(s^4Y(s)-s^3y(0)-s^2y'(0)-sy''(0)-y'''(0))=w_0e^{\frac{-LS}2$$

$$s^4Y(s)-sc_1-c_2=\frac{w_0}{EI}e^\frac{-LS}2$$

$$Y(s)=\frac{w_0}{EIs^4}e^{\frac{-LS}2}+\frac{c_1}{s^3}+\frac{c_2}{s^4}$$

$$y(x)=\frac{w_0}{6EI}(x-\frac{L}2)^3U(x-\frac{L}2)+\frac{c_1x^2}2+\frac{c_2x^3}6$$

$$y'(x)=\frac{w_0}{2EI}(x-\frac{L}2)^2U(x-\frac{L}2)+c_1x+\frac{c_2x^2}2$$

$$y''(x)=\frac{w_0}{EI}(x-\frac{L}2)U(x-\frac{L}2)+c_1+c_2x$$

$$y'''(x)=\frac{w_0}{EI}U(x-\frac{L}2)+c_2$$

$$y''(L)=\frac{Lw_0}{2EI}+c_1+c_2L=0$$

$$y'''(L)=\frac{w_0}{EI}+c_2=0$$

$$c_2=\frac{-w_0}{EI}$$

$$c_1=\frac{Lw_0}{2EI}$$

$$y(x)=\frac{w_0}{6EI}(x-\frac{L}2)^3U(x-\frac{L}2)+\frac{Lw_0x^2}{4EI}-\frac{w_0x^3}{6EI}$$

Last edited:
Glad to hear you got it solved. We love the easy ones

## 1. What is the Dirac delta function?

The Dirac delta function, also known as the Dirac delta distribution, is a mathematical function that is defined as zero for all values except at the origin, where it is infinite. It is commonly used in physics and engineering to represent point sources or idealized concentrated forces.

## 2. How is the Dirac delta function used in solving differential equations?

The Dirac delta function is often used in solving differential equations because it can be used to represent a forcing term or a boundary condition at a specific point. This allows for the solution of certain differential equations using the theory of distributions.

## 3. Can the Dirac delta function be integrated?

No, the Dirac delta function cannot be integrated in the traditional sense because it is not a typical function. However, it can be integrated in the context of the theory of distributions, where it is defined as the limit of a sequence of functions.

## 4. How does the Dirac delta function relate to the Kronecker delta function?

The Dirac delta function and the Kronecker delta function are both mathematical functions that are defined as zero for all values except at a specific point. However, the Dirac delta function is a continuous function, while the Kronecker delta function is a discrete function.

## 5. Are there any real-world applications of the Dirac delta function?

Yes, the Dirac delta function has many real-world applications, particularly in physics and engineering. It is used to model point sources of energy or mass, such as in the analysis of electric or gravitational fields. It is also used in signal processing to represent impulsive signals or sharp transitions.

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