# Differential Equation (not sure what order?)

In summary, the conversation involves discussing a given differential equation and how to solve it. The solution involves separating variables and obtaining y = ± √2 log(x) + c. The possibility of y = k being a solution is also discussed.

## Homework Statement

I don't think it's a typo since no one else in my class has asked about it, but the following DE was given on a handout:

$$x\frac{dy}{dx} - \frac{2}{x}\frac{dx}{dy} = 0$$

I am not sure how to deal with this one. Can I get a hint to get me started?

The order is the highest derivative which is one in this case. Also, with first-order derivatives, we can treat them like quotients and write:

$$xy'-\frac{2}{xy'}=0$$

Now, if you divide throughout by xy' then $xy'\ne 0$. Or if it is, then y'=0 or y=k which is one solution to the equation. To find the other one, divide by xy', take square root, keep track of plus and minus, then separate variables and integrate.

Ah... Thank you jackmell. I think I can take it from here.

Can I just multiply through by y' to get and divide by x to get

(y')2 - 2/x2 = 0

or

(y' - √2/x)(y' + √2/x) = 0

and then solve for each factor by separation?

Hmmm... then I will get as solution

y(x) = √2ln(x) - √2ln(x) + C1 + C2

y(x) = √2[ln(x) - ln(x)] +C1 + C2

y(x) = C1 + C2

This does not feel right ..

Last edited:
You have:

$$x^2(y')^2=2$$

or:

$$xy'=\pm \sqrt{2}$$

Now just separate variables and obtain:

$$y=\pm \sqrt{2}\log(x)+c$$

Also, I'm not so sure y=k is a solution since plugging that back into the DE causes it to be singular.

jackmell said:
You have:

$$x^2(y')^2=2$$

or:

$$xy'=\pm \sqrt{2}$$

Now just separate variables and obtain:

$$y=\pm \sqrt{2}\log(x)+c$$

Now note the solution y=k is not a particular case of those solutions however, I don't think it envelopes the general solution so y=k is not a singular solution but I'm not sure.

Hi jackmell So I got the same thing as you. However, isn't the general solution the superposition of the individual solutions? Hence when you add them, the solution reduces to a constant, i.e. C1 + C2 = k.

I am not sure how the solution is supposed to presented ... I feel like there should be a way around the log's dropping out ...

You mean a higher-ordered linear differential equation is the sum of a set of linearly independent solutions. For this equation however, the solution is simply:

$$y=c\pm \sqrt{2}\log(x)$$

It's easy to back-substitute either one of those and confirm it satisfies the DE.

## 1. What is a differential equation?

A differential equation is an equation that involves an unknown function and its derivatives. It describes the relationship between the function and its derivatives, and is used to model various natural and physical phenomena in science and mathematics.

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

An ordinary differential equation (ODE) involves only one independent variable, while a partial differential equation (PDE) involves multiple independent variables. ODEs are often used to model systems with a single variable, while PDEs are used to model systems with multiple variables, such as in physics and engineering.

## 3. How are differential equations solved?

Differential equations can be solved using analytical or numerical methods. Analytical methods involve finding a closed-form solution using mathematical techniques, while numerical methods involve approximating the solution using algorithms and computers.

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

Differential equations are used in various fields, such as physics, engineering, biology, economics, and finance. Some common applications include modeling population growth, predicting the spread of diseases, analyzing electrical circuits, and understanding the behavior of fluids and gases.

## 5. Are there any software programs that can solve differential equations?

Yes, there are many software programs available that can solve differential equations, such as MATLAB, Mathematica, and Maple. These programs use numerical methods to approximate the solutions to differential equations and are widely used in scientific research and engineering.

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