# Solving for a Differential Equation with Inspection.

• Kiziaru
In summary, the person is stuck on a problem and is looking for help. They mention that they have always lurked on this site, and that they don't know how to make the terms with m and n into exponents. They also mention that they have a vague idea, but that it doesn't involve a specific plan.
Kiziaru
I'm rather bad at solving by inspection, and right now I am stuck on a problem I can't solve.1.http://www.texify.com/img/%5CLARGE%5C%21y%28x%5E2y%5E2-m%29dx%2Bx%28x%5E2y%5E2%2Bn%29dy%3D0.gif
2.The attempt at a solution
http://www.texify.com/img/%5CLARGE%5C%21x%5E2y%5E3dx-mdx%2Bx%5E3y%5E2dy%2Bndy%3D0.gif

Rearrange and divide by x^2 * y^2

http://www.texify.com/img/%5CLARGE%5C%21ydx%2Bxdy%20%3D%20%28mydx-nxdy%29/x%5E2y%5E2.gif

I know that xdy+ydx = d(xy) buy I don't know what to do with the other side. If I try to make the right side into d(mx/ny) I'm left over with an x^2 that I don't know how to get rid of.

Last edited by a moderator:
Welcome to PF!

Hi Kiziaru! Welcome to PF!

(I haven't actually tried to solve this , but …)

my immediate reaction on seeing that m and n was to think "xm and yn"

tiny-tim said:
Hi Kiziaru! Welcome to PF!

(I haven't actually tried to solve this , but …)

my immediate reaction on seeing that m and n was to think "xm and yn"

Thanks! I've always lurked on this site, but I never posted because most of what I needed help with was already answered, until today.

Also, I hate to sound dumb, but what wizardry is this? How can I make m and n into exponents? And is this part of a plan to get d(arctan(y/x))?

Kiziaru said:
How can I make m and n into exponents? And is this part of a plan to get d(arctan(y/x))?

no specific plan … just a vague idea

remember, you can multiply the whole equation by anything you like

What I see is that if you ignore the terms with m and n, the other part is exact...

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with one or more of its derivatives. It describes the rate of change of a system or process and is often used in physics, engineering, and other scientific fields.

## 2. What is inspection method in solving differential equations?

The inspection method is a technique used to solve certain types of differential equations by guessing the form of the solution. It involves substituting the guessed solution into the differential equation and then solving for the unknown coefficients.

## 3. When is the inspection method useful in solving differential equations?

The inspection method is useful for solving differential equations that have simple and specific forms, such as linear, separable, and homogeneous equations. It is also helpful when the differential equation has constant coefficients.

## 4. What are the steps involved in using the inspection method to solve a differential equation?

The steps involved in using the inspection method are as follows:

1. Identify the type of differential equation and determine the form of the solution.
2. Guess the form of the solution and substitute it into the differential equation.
3. Solve for the unknown coefficients by equating the coefficients of the solution and the corresponding derivatives.
4. Verify the solution by substituting it back into the original differential equation.

## 5. What are some limitations of the inspection method in solving differential equations?

The inspection method may not work for all types of differential equations, particularly those with non-constant coefficients or non-linear forms. It also relies heavily on the ability to guess the correct form of the solution, which may not always be possible. In addition, it may not provide a general solution for the differential equation and may only give a specific solution for the given initial or boundary conditions.

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