# Solving a Simple Differential Equation: Comparing Different Approaches

• zezima1
In summary: We can write this in terms of a ratio of volumes too,$$\Rightarrow V_f=V_i\frac{T_f}{T_i}\text{.}In summary, the conversation discusses different methods of solving a differential equation, one using separation and the other using integration. Both methods lead to the same solution, but the integration method uses definite boundary conditions to solve for the constant. The conversation also shows how to arrive at the same solution using the separation method and how to find the constant using initial conditions. zezima1 So I have the differential equation: dV/dT = V/T I solve it with separation and get: ln(V) = ln(T) + c where c has to be figured out from initial conditions. Now this is how I am used to solving the equation. My book though does it differently. It simply integrate both sides from Vi to Vf or Ti to Tf. Why are these approaches the same? And how do you show that from the method I use? When they put definite boundary conditions on the integral, the constant was naturally solved for. Here's how you would go about it from the other method you mentioned. I'm going to assume the exercise in the book is using T for temperature and V for volume, and I will refer to them as such, but replace my words as necessary.$$
\int \frac{\mathrm{d}V}{V}=\int \frac{\mathrm{d}T}{T}\\
\Rightarrow \ln(V)=\ln(T)+C_0\text{, where }C_0\text{ is just some arbitrary constant.}\\
\Rightarrow V=e^{\ln(T)+C_0}=Te^{C_0}\equiv C_1 T\text{, where }C_1\text{ is some new arbitrary constant.}
$$Now we have$$
V=C_1T\text{.}
$$We assume that at some initial temperature, there will be some initial volume.$$
\Rightarrow V_i=C_1 T_i
\Rightarrow C_1=\frac{V_i}{T_i}
\Rightarrow V=\frac{V_i}{T_i}T
$$This equation will have to hold for the final temperature and volume as well,$$
\Rightarrow \frac{V_f}{T_f}=\frac{V_i}{T_i}


## 1. What is a simple differential equation?

A simple differential equation is an equation that involves a function and its derivatives. It is used to describe the relationship between a function and its rate of change over time or space.

## 2. Why is solving a differential equation important in science?

Solving differential equations is important in science because it allows us to model and understand complex systems and phenomena. Many natural and physical processes can be described using differential equations, making them essential in fields such as physics, engineering, and economics.

## 3. What are the different approaches to solving a differential equation?

There are several approaches to solving a differential equation, including analytical, numerical, and graphical methods. Analytical methods involve finding an exact solution using mathematical techniques, while numerical methods use algorithms to approximate the solution. Graphical methods involve plotting the function and its derivatives to visualize the behavior of the system.

## 4. How do I know which approach to use when solving a differential equation?

The approach to use when solving a differential equation depends on the specific equation and the information available. Analytical methods are preferred when an exact solution is needed, while numerical methods are useful for complex equations or when an exact solution is not possible. Graphical methods can be used to gain insight into the behavior of the system.

## 5. What are some real-world applications of solving differential equations?

Solving differential equations has many real-world applications, including predicting population growth, modeling the spread of diseases, understanding the behavior of electrical circuits, and optimizing chemical reactions. They are also used in fields such as economics, biology, and meteorology to study and predict various phenomena.

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