# First order differentials equations

• phospho
In summary, to transform the given homogeneous differential equation into a differential equation in z and x, we use z = y/x and solve the transformed equation to find the general solution of the original equation, giving y in terms of x. The steps involve simplifying the equation and separating the variables on either side to solve for y. The variable of integration is dz, and using dx instead will result in an incorrect solution.
phospho
Using z = y/x to transform the given homogeneous differential equation into a differential equation in z and x. By first solving the transformed equation, find the general solution of the original equation, giving y in terms of x.

$z = \frac{y}{x} \rightarrow y = xz \rightarrow \frac{dy}{dx} = x\frac{dz}{dx} + z$
$x\dfrac{dz}{dx} + z = \dfrac{x^3 + 4y^3}{3xy^2}$
$x\dfrac{dz}{dx} + z = \dfrac{x^3 + 4z^3x^3}{3xz^2x^2}$
$\dfrac{d}{dx}(xz) = \dfrac{1 + 4z^3}{3z^2}$
$xz = \displaystyle\int (\dfrac{1 + 4z^3}{3z^2} )$
$xz = \frac{-1}{3z} + \frac{2}{3}z^2 + c$
subbing z = y/x
$x = \frac{-x^2}{3y^2} + \frac{2y}{3x} + \frac{cx}{y}$

How do I rearrange from y from here? I just can't seem to do it :\

Last edited:
phospho said:
Using z = y/x to transform the given homogeneous differential equation into a differential equation in z and x. By first solving the transformed equation, find the general solution of the original equation, giving y in terms of x.

$z = \frac{y}{x} \rightarrow y = xz \rightarrow \frac{dy}{dx} = x\frac{dz}{dx} + z$
$x\dfrac{dz}{dx} + z = \dfrac{x^3 + 4y^3}{3xy^2}$
$x\dfrac{dz}{dx} + z = \dfrac{x^3 + 4z^3x^3}{3xz^2x^2}$
In the next step, instead of writing the left side as d/dx(xz), subtract z from both sides. Then you will have ## x \frac{dz}{dx}## on one side and whatever you get on the right side. At this point you can separate the equation so that one side has all the stuff involving x and dx, and the other has all the stuff involving z and dz.
phospho said:
$\dfrac{d}{dx}(xz) = \dfrac{1 + 4z^3}{3z^2}$
The step below is flaky. What is the variable of integration - dx or dz?
phospho said:
$xz = \displaystyle\int (\dfrac{1 + 4z^3}{3z^2} )$
$xz = \frac{-1}{3z} + \frac{2}{3}z^2 + c$
subbing z = y/x
$x = \frac{-x^2}{3y^2} + \frac{2y}{3x} + \frac{cx}{y}$

How do I rearrange from y from here? I just can't seem to do it :\

Mark44 said:
In the next step, instead of writing the left side as d/dx(xz), subtract z from both sides. Then you will have ## x \frac{dz}{dx}## on one side and whatever you get on the right side. At this point you can separate the equation so that one side has all the stuff involving x and dx, and the other has all the stuff involving z and dz.
The step below is flaky. What is the variable of integration - dx or dz?

I seem to get the right answer by separating the variables as you mentioned.

The variable of integration is dz (wont let me edit now...), is there an error in my original method?

phospho said:
I seem to get the right answer by separating the variables as you mentioned.

The variable of integration is dz (wont let me edit now...), is there an error in my original method?
It doesn't seem right time.

Starting from this step:
$$d/dx(xz) = \frac{1 + 4z^3}{3z^2}$$

What you're saying is that the derivative of xz with respect to x is <stuff on right side>, so xz must be the antiderivative of <stuff on right side> with respect to x. You can't just stick a dz in there out of nowhere.

## What is a first order differential equation?

A first order differential equation is a mathematical equation that involves a function and its derivative. It is typically written in the form dy/dx = f(x,y), where y is the dependent variable and x is the independent variable.

## What is the purpose of solving first order differential equations?

The main purpose of solving first order differential equations is to find the general solution that satisfies the equation for all possible values of the independent variable. This allows us to make predictions and analyze the behavior of systems in various fields such as physics, engineering, and economics.

## What are the different methods for solving first order differential equations?

Some common methods for solving first order differential equations include separation of variables, substitution, and using integrating factors. Other techniques such as exact equations, homogeneous equations, and Bernoulli equations can also be used depending on the specific form of the equation.

## What are the applications of first order differential equations in real life?

First order differential equations are used in various fields such as physics, engineering, economics, and biology to model and analyze the behavior of systems. They are also important in predicting and understanding natural phenomena such as population growth, radioactive decay, and chemical reactions.

## How can I check the validity of a solution to a first order differential equation?

To check the validity of a solution to a first order differential equation, you can substitute the solution into the equation and see if it satisfies the equation for all values of the independent variable. You can also use initial conditions or boundary conditions to verify the solution.

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