# How do i solve this differential equation

• seto6
In summary, the conversation discusses methods for solving a non-linear differential equation and recommends using the quadratic formula to solve for y'. The final equation reduces to y'= 2 or y'= 3y/x, depending on the chosen value in the quadratic formula.
seto6

## Homework Statement

x(y')2-(2x+3y)(y')+6y=0

non

## The Attempt at a Solution

i'm kinda lost on how to approch meaning what method to use..

all i know is that its a non linear differential equation...

Whenever you want to solve a non-linear one, try all the simple approaches in the chapter on non-linear equations in any DE textbook. One of those will work here. Get a DE textbook, find that chapter, then go over the example on eliminating the dependent variable. The first step is to solve for y in the equation:

$$x(y')^2-(2x+3y)y'+6y=0$$

Last edited:
Perhaps jackmell meant "The first step is to solve for y' ". That is what I recommend.

That is a quadratic equation in y'. You start by solving for y' using the quadratic formula:
$$y'= \frac{2x+ 3y\pm\sqrt{(2x+3y)^2- 24xy}}{2x}$$

It helps to note that $(2x+ 3y)^2- 24y= 4x^2+ 12xy+ 9y^2- 24xy$$= 4x^2- 12xy+ 9y^2= (2x- 3y)^2$!

The differential equation reduces to $y'= 2$ (using the "+") or $y'= 3y/x$ (using the "-").

HallsofIvy said:
Perhaps jackmell meant "The first step is to solve for y' ". That is what I recommend.

That is a quadratic equation in y'. You start by solving for y' using the quadratic formula:
$$y'= \frac{2x+ 3y\pm\sqrt{(2x+3y)^2- 24xy}}{2x}$$

It helps to note that $(2x+ 3y)^2- 24y= 4x^2+ 12xy+ 9y^2- 24xy$$= 4x^2- 12xy+ 9y^2= (2x- 3y)^2$!

The differential equation reduces to $y'= 2$ (using the "+") or $y'= 3y/x$ (using the "-").

No, I meant y using that method but your way was much simpler. :)

i see i thought of quadratic formula them said nah don't know why now i see how it works out!

## 1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between an unknown function and its derivatives. It involves one or more variables and their rates of change.

## 2. How do I know if a problem can be solved using a differential equation?

If a problem involves a relationship between a function and its derivatives, then it can be solved using a differential equation. Examples of problems that can be solved using differential equations include population growth, heat transfer, and motion of objects.

## 3. What are the different methods for solving a differential equation?

There are several methods for solving a differential equation, including separation of variables, Euler's method, and power series method. The choice of method depends on the type and complexity of the equation.

## 4. How do I solve a differential equation step by step?

The steps for solving a differential equation may vary depending on the method used. However, in general, the steps involve identifying the type of equation, finding the general solution, and applying initial or boundary conditions to find a particular solution.

## 5. Are there any software or tools that can help me solve a differential equation?

Yes, there are many software and tools available that can help you solve a differential equation. Some commonly used tools include MATLAB, Mathematica, and Wolfram Alpha. These tools use numerical and analytical methods to solve differential equations efficiently.

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