# Separable differential equations

• hitemup
In summary, the conversation discusses the manipulation of equations involving absolute values in order to solve for a constant. The process involves taking the natural log of both sides, simplifying, and then exponentiating to remove the ln. This simplification allows for the elimination of absolute values and the solving for the constant.
hitemup

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## The Attempt at a Solution

I've highlighted two equations on the screenshot. How did it proceed from the first to the second? I'm actually confused with the absolute values. What is the idea behind getting rid of the first absolute value(1-5v^2) while keeping the second one(x)?

So for starters, the manipulation is something like:
##-\frac 15 \ln | 1 - 5v^2 | = \ln |x| + c ##
##\begin{align*}
\ln | 1 - 5v^2 | &= -5(\ln |x| +c)\\
&= -5( \ln |x| - \ln e^{c} )\\
&=-5 ( \ln \frac{|x|}{e^{c} })\\
&= \ln \left(\frac{|x|}{e^{c} }\right)^{-5} \\
&= \ln \left(\frac{e^{5c} }{|x|^5}\right) \end{align*}##
Then, removing the absolute value from the left gives: ##1-5v^2 = \pm \left(\frac{e^{5c} }{|x|^5} \right) ##
So ##C = \pm e^{5c} ##
There only reason not to take out the absolute value from x that I can see is so that C is not dependent on x.

You get rid of the ln by exponentiating both sides.

## What are separable differential equations?

Separable differential equations are a type of differential equation in which the variables can be separated into two separate functions, one for each variable. This allows for easier integration and solution of the equation.

## How are separable differential equations solved?

To solve a separable differential equation, one must first separate the variables and then integrate both sides of the equation. This typically involves using techniques such as u-substitution or integration by parts.

## What is the purpose of solving separable differential equations?

Solving separable differential equations allows us to model and understand real-world phenomena, such as growth or decay rates, population dynamics, and chemical reactions. It also plays a crucial role in many areas of science and engineering.

## What is the difference between separable and non-separable differential equations?

The main difference between separable and non-separable differential equations is that separable equations can be written as a product of two separate functions, while non-separable equations cannot be written in this form. Non-separable equations often require more advanced techniques to solve.

## What are some common applications of separable differential equations?

Separable differential equations are commonly used in physics, engineering, and other fields to model and understand various physical phenomena. They are also used in economics and finance to model growth and decay rates. Additionally, they are used in biology to model population dynamics and in chemistry to model reaction rates.

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