# Nonlinear Differential Equation Homework Attempt

• Fizz_Geek
In summary, the conversation is about solving the equation r''=1/r^2 and the suggested approach is to multiply both sides by r' and integrate. The conversation also includes a discussion on the next steps of integration and the usefulness of the trick.

r" = 1/r^2

## Homework Equations

A friend gave this to me, he was just wondering how we'd approach it.

## The Attempt at a Solution

I don't think the equation is linear, so I don't know how to approach it. My friend suggested integrating both sides with respect to r, but we don't know if that's legal because we don't know what r might be a function of.

Multiple both sides by r' and that will give you the first integral.

hunt_mat said:
Multiple both sides by r' and that will give you the first integral.

How's does multiplying by a derivative give me an integral?

As
$$r''=\frac{1}{r^{2}}$$
Multiply both sides by r' to obtain:
$$r'r''=\frac{r'}{r^{2}}\Rightarrow\left(\frac{(r')^{2}}{2}\right) '=\left( -\frac{1}{r}\right) '$$
Integrate easily from here.

Did you try it? What is the derivative of $(r')^2$? So what is the integral of $r' r''$? What is the integral of $dr/r^2$?

To hunt_mat:

That's a very nice trick! I got stuck on the next step, integrating (r')2, but I'll try to work that out with my friend before asking again.

Thanks very much!

It's a standard trick, become familier with it.

## 1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives to represent rates of change and is commonly used to model real-world phenomena in physics, engineering, and other fields.

## 2. What is the difference between an ordinary and a partial differential equation?

An ordinary differential equation (ODE) involves a single independent variable and its derivatives, while a partial differential equation (PDE) involves multiple independent variables and their partial derivatives. ODEs are used to model phenomena that change over time, while PDEs are used to model phenomena that vary over space and time.

## 3. What are the applications of differential equations?

Differential equations have a wide range of applications in various fields, including physics, engineering, biology, economics, and finance. They are used to model and analyze complex systems and phenomena, such as population growth, heat transfer, fluid dynamics, and electrical circuits.

## 4. How do you solve a differential equation?

There are various methods for solving differential equations, depending on the type of equation and its complexity. Some common techniques include separation of variables, substitution, and using integrating factors. In some cases, differential equations can be solved numerically using computer software.

## 5. What are the initial and boundary conditions in a differential equation?

The initial conditions specify the values of the dependent variable and its derivatives at a specific initial value of the independent variable. They are used to determine the particular solution of a differential equation. Boundary conditions, on the other hand, are used to determine the general solution of a differential equation by specifying the values of the dependent variable at the boundaries of the domain.