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Differential Equations - Introduction
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Never miss a course! It has only the first derivative dy dx , so is "First Order". This has a second derivative d 2 y dx 2 , so is "Order 2". This has a third derivative d 3 y dx 3 which outranks the dy dx , so is "Order 3". The degree is the exponent of the highest derivative.
Be careful not to confuse order with degree. Some people use the word order when they mean degree! It is Linear when the variable and its derivatives has no exponent or other function put on it. This is not a complete list of how to solve differential equations, but it should get you started:. Hide Ads About Ads. But first: why? Why Are Differential Equations Useful? In our world things change, and describing how they change often ends up as a Differential Equation: Example: Rabbits!
The important parts of this are: the population N at any time t the growth rate r the population's rate of change N Let us imagine some actual values: the population N is the growth rate r is 0. Remember: the bigger the population, the more new rabbits we get! Example: Compound Interest Money earns interest. This is called compound interest.
Earl Coddington - An introduction to Ordinary Differential Equations.pdf
And the bigger the loan the more interest it earns. Solving The Differential Equation says it well, but is hard to use. Example: Rabbits Again! Example: Spring and Weight A spring gets a weight attached to it: the weight gets pulled down due to gravity, as the spring stretches its tension increases, the weight slows down, then the spring's tension pulls it back up, then it falls back down, up and down, again and again. We also show who to construct a series solution for a differential equation about an ordinary point. The method illustrated in this section is useful in solving, or at least getting an approximation of the solution, differential equations with coefficients that are not constant.
Note that while this does not involve a series solution it is included in the series solution chapter because it illustrates how to get a solution to at least one type of differential equation at a singular point. Linear Homogeneous Differential Equations — In this section we will extend the ideas behind solving 2 nd order, linear, homogeneous differential equations to higher order. We will also need to discuss how to deal with repeated complex roots, which are now a possibility. In addition, we will see that the main difficulty in the higher order cases is simply finding all the roots of the characteristic polynomial.
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Undetermined Coefficients — In this section we work a quick example to illustrate that using undetermined coefficients on higher order differential equations is no different that when we used it on 2 nd order differential equations with only one small natural extension. Variation of Parameters — In this section we will give a detailed discussion of the process for using variation of parameters for higher order differential equations.
We will also develop a formula that can be used in these cases. We will also see that the work involved in using variation of parameters on higher order differential equations can be quite involved on occasion.
Laplace Transforms — In this section we will work a quick example using Laplace transforms to solve a differential equation on a 3 rd order differential equation just to say that we looked at one with order higher than 2 nd. As we will see they are mostly just natural extensions of what we already know who to do.
Series Solutions — In this section we are going to work a quick example illustrating that the process of finding series solutions for higher order differential equations is pretty much the same as that used on 2 nd order differential equations. We will also work a few examples illustrating some of the interesting differences in using boundary values instead of initial conditions in solving differential equations. Eigenvalues and Eigenfunctions — In this section we will define eigenvalues and eigenfunctions for boundary value problems.
We will work quite a few examples illustrating how to find eigenvalues and eigenfunctions. In one example the best we will be able to do is estimate the eigenvalues as that is something that will happen on a fairly regular basis with these kinds of problems. Periodic Functions and Orthogonal Functions — In this section we will define periodic functions, orthogonal functions and mutually orthogonal functions. The results of these examples will be very useful for the rest of this chapter and most of the next chapter.
We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function.
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We will also define the even extension for a function and work several examples finding the Fourier Cosine Series for a function. Fourier Series — In this section we define the Fourier Series, i. We will also work several examples finding the Fourier Series for a function. Convergence of Fourier Series — In this section we will define piecewise smooth functions and the periodic extension of a function. In addition, we will give a variety of facts about just what a Fourier series will converge to and when we can expect the derivative or integral of a Fourier series to converge to the derivative or integral of the function it represents.
The Heat Equation — In this section we will do a partial derivation of the heat equation that can be solved to give the temperature in a one dimensional bar of length L. In addition, we give several possible boundary conditions that can be used in this situation. We also define the Laplacian in this section and give a version of the heat equation for two or three dimensional situations.
The Wave Equation — In this section we do a partial derivation of the wave equation which can be used to find the one dimensional displacement of a vibrating string. In addition, we also give the two and three dimensional version of the wave equation.
Terminology — In this section we take a quick look at some of the terminology we will be using in the rest of this chapter. In particular we will define a linear operator, a linear partial differential equation and a homogeneous partial differential equation. We also give a quick reminder of the Principle of Superposition.
Separation of Variables — In this section show how the method of Separation of Variables can be applied to a partial differential equation to reduce the partial differential equation down to two ordinary differential equations. We apply the method to several partial differential equations. We do not, however, go any farther in the solution process for the partial differential equations. That will be done in later sections. The point of this section is only to illustrate how the method works. Solving the Heat Equation — In this section we go through the complete separation of variables process, including solving the two ordinary differential equations the process generates.
We will do this by solving the heat equation with three different sets of boundary conditions. Heat Equation with Non-Zero Temperature Boundaries — In this section we take a quick look at solving the heat equation in which the boundary conditions are fixed, non-zero temperature. Note that this is in contrast to the previous section when we generally required the boundary conditions to be both fixed and zero.
As we will see this is exactly the equation we would need to solve if we were looking to find the equilibrium solution i. Vibrating String — In this section we solve the one dimensional wave equation to get the displacement of a vibrating string. Summary of Separation of Variables — In this final section we give a quick summary of the method of separation of variables for solving partial differential equations.