# I need a little help with differentials

• James2
In summary: So, in summary, you need to learn integration before you can solve differential equations. And even then, there are some equations that are just too hard to solve with the techniques we have at our disposal. So, we need to understand the theory that makes getting a solution actually work in the first place.
James2
Alright, I know my algebra, geometry, and plane trigonometry. I've been calculating derivatives for a while now, but I wish to get to a higher level of differentiation. Say, differential equations. Could somebody explain how to solve a differential equation?

Welcome to PF!

Hi James2! Welcome to PF!
James2 said:
Alright, I know my algebra, geometry, and plane trigonometry.

Sorry, but you need to know integration before you can solve differential equations.

(basically, because you have to "undo" the equation, and "undoing" a derivative is integration)

here's some you can do:solve y' - cy = 0. If you have been differentiating much you know that e^ct works.So here is another similar one:

solve y'' - 3y' + 2y = 0.

to solve it just guess the solution to be e^ct, and differentiate to find c.

You will see that e^ct works if and only if c solves the polynomial equation

x^2 -3x + 2 = 0, i.e. if and only if c = 1,2.what about y'' + y' -2y = 0?

James2 said:
Alright, I know my algebra, geometry, and plane trigonometry. I've been calculating derivatives for a while now, but I wish to get to a higher level of differentiation. Say, differential equations. Could somebody explain how to solve a differential equation?

Hey James2 and welcome to the forums.

This question that is seemingly so simple, is actually not an easy one to answer.

The thing is that not all differential equations were created equal.

In some cases you can use some rather simple substitutions or calculus identities (like the chain rule) to get the answer and for others, you have to resort to things like integral transforms like the Laplace transform.

Even then with the mathematical machinery that we have, many differential equations like non-linear ones just can't be solved with the techniques we have at the moment and because of this we need to not only use computers, but we have to understand the theory that makes getting a solution actually work to begin with.

So keep the above in mind: in other words, we can't always solve a DE like we solve say a set of equations like x + 2y = 4 and x + 3y = 7 where we get x = blah and y = blah: it's not that easy when you start getting to the complicated kind of equations.

In terms of understanding differentiation though, once you understand all the basics like the chain rule, product rule, quotient rule and others including partial derivatives, then the rest is pretty much using these rules in a given context.

Sure, I would be happy to help with differentials and differential equations. First, let's define what a differential is. A differential is a mathematical concept that represents the instantaneous rate of change of a function. It is denoted by dy/dx or f'(x), where y is the dependent variable and x is the independent variable.

To solve a differential equation, you will need to use techniques from calculus, specifically integration and differentiation. There are different types of differential equations, such as ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve only one independent variable, while PDEs involve multiple independent variables.

To solve an ODE, you can use techniques such as separation of variables, substitution, or integrating factors. These methods involve manipulating the equation to isolate the dependent variable and then integrating both sides to find the solution. It is helpful to have a good understanding of algebra, geometry, and trigonometry to solve ODEs.

For PDEs, the process is similar, but it may involve more advanced techniques such as Fourier series, Laplace transforms, or the method of characteristics. These methods can be more complex and require a deeper understanding of calculus and linear algebra.

In addition to understanding the mathematical techniques, it is important to have a good conceptual understanding of differential equations. This involves understanding the physical or real-world meaning of the equation and how it relates to the system or phenomenon being studied.

I hope this explanation helps you understand how to solve differential equations. If you have any specific questions or need further clarification, please let me know.

## 1. What are differentials?

Differentials are mathematical tools used to calculate the rate of change of a function. They are used to find the instantaneous rate of change at a specific point on a curve or surface.

## 2. How are differentials used in real life?

Differentials have many applications in real life, such as in physics, engineering, economics, and biology. They are used to model and predict changes in a system over time, such as in the stock market or in population growth.

## 3. What is the difference between a differential and a derivative?

A differential is the result of a derivative operation, while a derivative is the rate of change of a function. In other words, a differential represents the change in a function, while a derivative represents the ratio of change.

## 4. How do I solve problems involving differentials?

To solve problems involving differentials, you need to first understand the concept of derivatives and how to take them. Then, you can use the rules of differentiation to find the differential of a function. Finally, you can substitute the given values into the differential equation and solve for the unknown variable.

## 5. Can I approximate a function using differentials?

Yes, you can use differentials to approximate a function. This is known as linear approximation or tangent line approximation. It involves using the tangent line at a specific point on a curve to estimate the value of the function at that point.

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