Welcome to Differential Equations at 17Calculus. Differential Equations is a vast and incredibly fascinating topic that uses calculus extensively. This page gets you started on Ordinary/Elementary Differential Equations usually covered in a first semester differential equations course.
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What Are Differential Equations?
Differential Equations consists of a group of techniques used to solve equations that contain derivatives. That's it. That's all there is to it. The complexity comes in 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 one 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 to teach. 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 |
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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 |
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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 it is to read the exponent? We will follow this convention on this site.
Notation For Derivatives |
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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 (two dots for the second derivative). |
Getting Started
After going through the above information you are ready to watch some videos to get started with differential equations.
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. He also works the example \( y'' + 2y' - 3y = 0 \) and shows that \( y_1 = e^{-3x} \) and \( y_2 = e^x \) are solutions to this differential equation. Then, he goes on to explain linear versus nonlinear and order.
video by Khan Academy |
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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.
video by Khan Academy |
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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.
video by PatrickJMT |
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Okay, now that you have an overview, you need to learn how to classify differential equations. This is important since the techniques that you will learn apply only to certain types of equations.
You CAN Ace Differential Equations
external links you may find helpful |
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The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1 - basic identities | |||
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\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) |
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) |
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) |
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) |
Set 2 - squared identities | ||
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\( \sin^2t + \cos^2t = 1\) |
\( 1 + \tan^2t = \sec^2t\) |
\( 1 + \cot^2t = \csc^2t\) |
Set 3 - double-angle formulas | |
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\( \sin(2t) = 2\sin(t)\cos(t)\) |
\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\) |
Set 4 - half-angle formulas | |
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\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\) |
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) |
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) |
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\) | |
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) |
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\) | |
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) |
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\) |
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\) |
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\) | |
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) |
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\) | |
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\) |
Trig Integrals
\(\int{\sin(x)~dx} = -\cos(x)+C\) |
\(\int{\cos(x)~dx} = \sin(x)+C\) | |
\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\) |
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\) | |
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) |
\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\) |
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