# First order separable differential equation

• amolv06
In summary, the conversation discusses solving an equation involving dy/dx and x, where the attempt at a solution incorrectly uses t instead of x and suggests using partial fractions. However, it is pointed out that partial fractions are not needed and the correct solution involves finding y.
amolv06

## Homework Statement

$$\frac{dy}{dx} + x^{2} = x$$

Above.

## The Attempt at a Solution

After rearranging, I am stuck at

$$\int \frac{1}{x-x^{2}} dx = \int dt$$

I can't think of any u-substitution, or any other trick for integrals I could use to solve this.

Recheck you're expressions. The problem you give involves x and y, while the attempt at a solution gives x and t. Please clarify.

It looks like you should have dx/dt where you currently have dy/dx no? If that's the case consider a partial fraction decomposition.

I wrote the wrong variable for the last integral. It should have been of dy rather than dt. I solved it. I forgot about partial fractions.

Even so, $dy/dx= x- x^2$ becomes $dy= (x- x^2)dx$. There is no need for partial fractions.

What need to find if y, then
$$y=\int(x-x^2)dx=\frac{x^2}{2}-\frac{x^3}{3}+C$$

## What is a first order separable differential equation?

A first order separable differential equation is a type of differential equation where the variables can be separated into two separate functions. This means that the dependent variable can be expressed as a product of two functions, one of which contains only the independent variable and the other contains only the dependent variable.

## How do you solve a first order separable differential equation?

To solve a first order separable differential equation, you need to separate the variables and integrate both sides of the equation. This will result in two separate equations, one for each function. Then, you can solve each equation separately and combine the solutions to get the final answer.

## What is the importance of first order separable differential equations?

First order separable differential equations are important in many areas of science, including physics, chemistry, and engineering. They are used to describe natural phenomena and to make predictions about how systems will behave over time. They are also used in the development of mathematical models and simulations.

## What are some real-life applications of first order separable differential equations?

First order separable differential equations have many real-life applications, such as predicting population growth, modeling chemical reactions, and analyzing electrical circuits. They are also used in economics to model supply and demand and in biology to study population dynamics.

## What are some common techniques for solving first order separable differential equations?

Some common techniques for solving first order separable differential equations include separation of variables, substitution, and integrating factors. These techniques involve manipulating the equations to make them easier to solve and using integration methods to find the solutions. Other techniques, such as using differential equations software, may also be used.

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