# A linear differential equation problem

• A330NEO
In summary, the problem asks for a differential equation using Newton's second law to determine the height x(t) of one end of a uniform, 10-foot-long heavy rope being pulled vertically upward with a constant force of 5lb. The equation involves expressing the momentum of the rope in terms of x and its derivatives, and can be written as 5-x = x(d^2x/dt^2) - [(dx/dt)^2], where x represents position and t represents time. The left side of the equation may need to be multiplied by 32 (gravitational acceleration) to match the answer in the book.
A330NEO

## Homework Statement

A uniform 10-foot-long heavy rope is coiled loosely on the ground. One end of the rope is pulled vertically upward by means of a constant force of 5lb. The rope weighs 1lb/ft. Use Newton's second law to determine a differential equation for the height x(t) of the end above ground level at time t. Assume that a positive direction is upward.

## Homework Equations

The answer says it's $x \frac{d^{2}x}{dt^2} + \left ( \frac{dx}{dt} \right )^{2}+32x=160$

## The Attempt at a Solution

Since Newton's second law is F=ma, I tried this:
a(Acceleration) is position x differentiated twice, so $a=\frac{d^{2}x}{dt^2}$
m=x, and force is 5-x. so, the equation becomes
$5-x=x \frac{d^{2}x}{dt^2}$
is anything wrong?

Last edited:
Yes, you cannot use F = ma, which is only true when the mass in question is constant. You need to use the second law on the more general form F = dp/dt, where p is the momentum.

Orodruin said:
Yes, you cannot use F = ma, which is only true when the mass in question is constant. You need to use the second law on the more general form F = dp/dt, where p is the momentum.
may I ask you how to apply it?

Try expressing the momentum of the rope in terms of x and its derivatives.

Orodruin said:
Try expressing the momentum of the rope in terms of x and its derivatives.
you mean this equation? this is from Wikipedia, and it's based on momentum equation p=mv, thus F=dp/dt.
$F=m(t)\frac{dv}{dt} - u\frac{dm}{dt}$
When I put this I that question, I found something like this.
$5-x= x\frac{d^{2}x}{dt^{2}} - \left ( \frac{dx}{dt} \right )^2$
Looks similar to the 'answer' in my book, but still not getting it. I suspect that the left side of the equation have been multiplied by 32(Gravitational acceleration in English unit, I think) but hard to find where to put it.

Type your equation in this box

## 1. What is a linear differential equation?

A linear differential equation is a mathematical equation that involves an unknown function and its derivatives, where the function and its derivatives are only raised to the first power. It can be written in the form y' = f(x,y) or y'' = g(x,y), where y is the unknown function and f and g are functions of x and y.

## 2. What makes a differential equation linear?

A differential equation is linear if the unknown function and its derivatives are only raised to the first power and there are no products or powers of the function or its derivatives. This means that the equation can be written as a linear combination of the function and its derivatives.

## 3. What is the purpose of solving a linear differential equation problem?

The purpose of solving a linear differential equation problem is to find the unknown function that satisfies the equation. This can be useful in predicting the behavior of a system or understanding a physical phenomenon.

## 4. What are the steps for solving a linear differential equation problem?

The steps for solving a linear differential equation problem are: 1) write the equation in standard form, 2) determine the integrating factor, 3) solve for the unknown function, and 4) check the solution by plugging it back into the original equation to see if it satisfies the equation.

## 5. What are some real-world applications of linear differential equations?

Linear differential equations have many real-world applications in fields such as physics, engineering, biology, and economics. They can be used to model population growth, radioactive decay, electrical circuits, and many other phenomena that involve rates of change.

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