## Differential Equations

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Differential Equations is a vast and incredibly fascinating topic that uses calculus extensively. Although the name does not have the word 'calculus' in it, differential equations is a natural extension and application of calculus. So it is thoroughly covered on this site including discussion, videos and practice problems. This page gets you started on Ordinary Differential Equations usually covered in a first semester differential equations course. If you are looking for more detail on a specific differential equations topic, you can select a topic from the drop-down menu above or open this next panel of topic links for more targeted links to topics, videos and practice problems.

What Are Differential Equations and How Can You Prepare Yourself?

Differential Equations is a group of topics to teach you to solve equations that contain derivatives. That's it. That's all there is to it. The complexity happens because you can't just integrate the equation to solve it. First, you need to classify what kind of differential equation it is based on several criteria. Then, you can choose a technique to solve. Learning to solve differential equations involves learning to classify the equation you are given and then learning the technique to solve that specific type of equation. There is generally no technique that works in all cases. So, to prepare yourself, spend some extra effort learning to classify the kind of equation you have as you learn each technique. If you don't, you will be totally lost.

There are a lot of shortcuts to solving differential equations. Many instructors teach those shortcuts upfront precisely because they are easier. However, don't let yourself lose sight of where those shortcuts come from and under what conditions you can use them. Spend some time learning the basic technique before using the shortcuts. This usually involves working the first few practice problems with the basic technique. Of course, the instructions your teacher gives you take priority. But really learn these techniques, so that you will know the proper time and situation to use them. After all, that's the point, right? To be able to use this material in your job or other courses?

### Notation

Constants and Variables
One of the first things you need to get your head around with differential equations is which symbols are constants and which are variables. When you see derivative notation you will mostly see $$y'$$ instead of $$dy/dt$$, for example. So you need to keep track of which symbols are functions, which are variables, what you are taking the derivative with respect to and what are constants.

For example, one equation I ran across in the first section of a differential equations textbook was
$$\displaystyle{ m\frac{dv}{dt} = mg-\gamma v }$$
This could have been written $$\displaystyle{ mv' = mg-\gamma v }$$
In this case, the variable is $$t$$ and the function is $$v(t)$$. The symbols $$m, g, \gamma$$ are constants. The context of the problem is important to read and understand in order to arrive at these conclusions.

exp Notation
A second thing you need to be aware of is that some textbooks (most of the ones I've seen) use unusual notation for exponential functions. Correct notation is $$y = e^x$$. However, sometimes the exponent can be very long and contain a lot of detail. So, the exponential function will sometimes be written as $$y = exp(x)$$. This is only used when the exponent $$x$$ is detailed. For example, $$\displaystyle{ \mu(t) = e^{ \int_{t_0}^{t}{p(s)~ds} } }$$ is difficult to read. Since there is so much detail in the exponent of $$e$$ that we need to see, we usually write this
$$\displaystyle{ \mu(t) = exp\left( \int_{t_0}^{t}{p(s)~ds} \right) }$$
See how much easier the exponent is to read?
We will follow this convention on this site.

Notation For Derivatives
By now you should be comfortable with the notation $$dy/dx$$ and $$y'$$ for the first derivative. There are a couple of other types of notation that you may or may not have seen before, that you will probably run across on this site, in your textbook, in class and in videos.

 $$D_x(y)$$ where $$D$$ tells you take the derivative and the subscript $$x$$ is the variable of integration. $$\dot{y}$$ where the dot above the $$y$$ tells you to take the first derivative.

Some Definitions Used To Classify Differential Equations

### Order

Order indicates the highest derivative appearing in the differential equation. For example, $$y' - y = 0 ~$$ is a first order differential equation because the highest derivative is a first derivative. Similarly, $$y'' - y = 0$$ is a second order differential equation because the highest derivative is a second derivative.

### Linear

A differential equation of the form
$$P(t) y'' + Q(t)y' + R(t)y = G(t)$$
is linear since $$P(t)$$, $$Q(t)$$ and $$R(t)$$ are functions of $$t$$ only, i.e. they do not contain any $$y$$'s or derivatives of $$y$$. Analyzing nonlinear equations is relatively difficult, so it is unlikely you will encounter them in a first semester differential equations course, except under very special circumstances.

### Homogeneous

There are several meanings of the term 'homogeneous' used in differential equations. For this definition, assume you have a differential equation of the form $$p(t)y'' + q(t)y' + r(t)y = g(t)$$.
Meaning 1 - - If $$g(t) = 0$$ in the above form, then the differential equation is said to be homogeneous. The idea is to get all the terms containing $$y$$ or a derivative of $$y$$ to one side of the equal sign and all other terms to the other side. If there are no terms without a $$y$$ or it's derivatives, then $$g(t)$$ will be zero and the equation will be homogeneous. If $$g(t) \neq 0$$, then the differential equation is said to be inhomogeneous (or nonhomogeneous).
Meaning 2 - - If all of the expressions $$p(t)$$, $$q(t)$$, $$r(t)$$ and $$g(t)$$ can be written as functions of $$y/t$$, then it is said to be homogeneous. In this case, we use the technique of substitution to solve this type of differential equation.

Getting Started with Differential Equations

After going through the above information you are ready to watch some videos to get started on differential equations.
Video 1 - - Here is a good introduction to differential equations. He contrasts a differential equation to a standard equation, which you should be familiar with and explains, practically, what a differential equation is.   open video

Video 2 - - Here is another introduction video. The technique he uses is separation of variables, which is the first technique usually introduced in a differential equations course. It will help you to see this technique in the context of introducing differential equations.   open video

Video 3 - - Here is a good video showing what it means for an equation to be a solution to a differential equation. This also demonstrates how to check your answer after you have solved a differential equation.   open video

Next - - Now that you have an overview of differential equations, you are ready to begin studying specific topics. The next natural topic is slope fields. However, many instructors will go straight into separation of variables. You can also choose your own next topic by selecting from the list of topics from the menu at the top of the page. In any case, take some time to enjoy studying differential equations.

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### faq: What is the difference between a general solution and the particular solution?

#### What is the difference between a general solution and the particular solution?

Short Answer: The presence (general solution) or absence (particular solution) of an unknown constant.
When you solve a differential equation, you use integration, which introduces an unknown constant.
In a general solution, the unknown constant remains and you do not have enough information to be able to determine what that constant is. Consequently, you have an infinite number of solutions.
In the particular solution, you start out by solving for the general solution but then you are given initial conditions which you use to determine the value of the constant. These initial conditions are actually points that the solution to the differential equation pass through. In the end, you have only one solution without any unknown constants.
Notice that we say a general solution but we talk about the particular solution.

general solution particular solution infinite number of solutions only one solution contains unknown constant(s) does NOT contain any unknown constants no initial conditions given initial conditions given; used to solve for constants

To find a general solution, just solve the differential equation and leave any constants in your final answer.
To find the particular solution, find the general solution first, then plug in the initial conditions and solve for the constants.

Notes
1. When determining the particular solution, you will be given the same number of initial conditions as the order of the differential equation. Depending on how the initial conditions are given, you may need to stop after each integration and solve for the individual constants or you may need to wait until you are completely done and solve for all the constants at once.
2. When you were first learning integration, you probably ran across initial value problems. These were actually differential equations where you were asked to find the particular solution.