# Second order differential equation

• elevenb
In summary, the conversation is about a problem involving two particles connected by a spring with a spring constant k and zero equilibrium length. The link provides all necessary equations and the attempt at a solution involves using the Binomial theorem to solve for the angular frequency. The last step uses the standard solution for the differential equation of simple harmonic motion.
elevenb

## The Attempt at a Solution

The solution is there, its to do with part (c), I would really appreciate it if someone would explain how they make the jumps - especially to the second last equation

They used the Binomial theorem.
$$\frac{1}{(d + \Delta d)^2} =\frac{1}{d^2}\left(1 + \frac{\Delta d}{d}\right)^{-2} = \frac{1}{d^2}\left(1 - 2\frac{\Delta d}{d} + \cdots\right)$$

Everything before that looks like straightforward algebra. The last step to find ##\omega## is the standard solution of the differential equation for simple harmonic motion.

Ignore- spotted mistake

Last edited:

## What is a second order differential equation?

A second order differential equation is a mathematical equation that describes the relationship between a function and its second derivative. It is typically used to model physical phenomena in science and engineering.

## How is a second order differential equation different from a first order differential equation?

A second order differential equation involves the second derivative of the function, while a first order differential equation involves only the first derivative. This means that a second order differential equation requires two initial conditions to be fully determined, whereas a first order differential equation only requires one initial condition.

## What are some real-world applications of second order differential equations?

Second order differential equations can be used to model a variety of physical phenomena, such as the motion of a spring, the oscillations of a pendulum, and the flow of fluids. They are also commonly used in engineering to analyze and design systems such as electrical circuits and control systems.

## How do you solve a second order differential equation?

There are various methods for solving second order differential equations, including separation of variables, substitution, and using the characteristic equation. The specific method used will depend on the form of the equation and the initial conditions given.

## What are the initial conditions in a second order differential equation?

The initial conditions in a second order differential equation refer to the values of the function and its first derivative at a specific point. These values are necessary to fully determine the equation and find a unique solution.

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