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This course will teach everything that is usually taught in the first two semesters of a university/college course in differential equations. The topics we will consider in this course are
Here we look at a differential equation which models the motion of a falling object under the force of gravity.
Here we look at a differential equation which models the motion of a falling object under the force of gravity and air resistance.
Here we look at a differential equation which models the motion of mass on a spring.
Here we look at a differential equation which models RLC circuits. It is very interesting to find that the governing differential equation has the exact same form as the differential equation which models the motion of a mass on a spring.
Here we look at a differential equation which models the motion of a simple pendulum.
Here we look at several differential equations which occur in applications. We consider the equation for a hanging rope, Newtons law of cooling, the deflection of a cantilever beam, and a simple population growth model.
In this video we define what an ordinary differential equation is and also how to classify them in terms of their order and whether they are linear or nonlinear.
Here we define what it means for a differential equation to have a solution. We do some simple examples where we verify a function is a solution to a differential equation.
Here we show that solutions to differential equations can be explicit or implicit.
In this video we learn about slope fields and solution curves for first order differential equations.
In this video we learn about the existence and uniqueness theorem for first order differential equations.
In this video we define what separable differential equations are and learn how to solve them.
In this video we solve a separable differential equation. We also quickly review integration by parts.
In this video we discuss the differential equation for Newtons law of cooling and solve it. We can solve the equation because its separable.
In this video we learn how to find the time of death when a homicide victim is found in a room with some temperature. This is an application of Newtons law of cooling.
In this video we use Torricellis Law and calculus to derive the differential equation for water draining from a tank with a hole in its base.
Here we solve a differential equation for the height of water draining out of a conical water tank. Using our solution we can calculate how long it takes for the tank to drain.
In this video we learn to solve linear first order differential equations.
Here we derive a differential equation for simple mixing problems. Mixing problems are a nice application of first order linear differential equations. In these problems a solute/solvent mixture is added to a tank with a similar mixture. The mixture is then pumped out of the tank. A differential equation for the amount of solute in the tank is derived.
In this video we do an example of a mixing problem with one tank.
In this video we learn how to solve exact differential equations.
In this video we go through an example of solving an exact differential equation.
In this video we go through another example of solving an exact differential equation.
In this video we begin to look at substitution methods. In particular we learn how to solve differential equations of the form dy/dx=f(ax+by+c) by making an appropriate substitution.
In this video we define what first order homogenous differential equations are. We also define homogeneous functions. These homogeneous differential equations can be solved by a substitution. This video shows how this is done.
In this video we go through an example of solving a first order homogeneous differential equation.
In this video we go through a second example of solving a first order homogeneous differential equation.
In this video we learn how to solve Bernoulli Differential Equations. They can be solved by an appropriate substitution.
In this video we see how substitutions can sometimes reduce second order differential equations to first order differential equations which we can then solve.
This assignment tests your understanding of the material presented in Sections 1 and 2.
In this video we begin to look at higher order differential equations. In particular we focus on the theory of linear seconder order differential equations. We define a differential operator that we will be using through the next several videos. We also define homogeneous second order equations and non-homogeneous equations.
In this video we define what a linear operator is and discuss how the 2nd order differential operator defined in the previous video is a linear operator.
In this video we look at an important theorem known as the principal of superposition for linear homogeneous equations. It says that if y1 and y2 are two solutions to a homogeneous equation, then so is any linear combination of them.
In this video we state the existence and uniqueness theorem for higher order differential equations.
In this video we define the Wronksian determinant and see its importance in the theory of linear differential equations. In particular, if y1 and y2 are two different solutions to a second order linear homogeneous DE, then constants c1 and c2 can be chosen to solve an IVP provided that the Wronskian of y1 and y2 is non zero.
This video continues from the previous one. We state a main theorem summarizing the information from the last video while also adding another important statement. We discuss how every solution to linear homogeneous differential equations can be written in a certain way.
In this video we summarize the main points of interest regarding the solutions of linear second order homogeneous equations.
Here we introduce the idea of linear independence and dependence for two functions and relate it to the Wronskian of two functions.
In this video we look at a stronger theorem relating linear independence/dependence to the wronskian for solutions of second order linear homogenous equations.
In this video we discuss the theory of nth order linear homogeneous equations. We learn about linear independence/dependence and the wronskian for n>2 functions. We also discuss whats needed for general solutions. The theory is just an extension of the theory for 2nd order equations we have seen in the last several videos.
In this video we begin to learn how to solve second order homogeneous equations with constant coefficients.
In this video we learn how to solve second order homogeneous equations with constant coefficients when the roots of the characteristic equation are real and distinct.
In this video we learn how to solve second order homogeneous equations with constant coefficients when there is only one real root of the characteristic equation.
In this video we learn how to solve second order homogeneous equations with constant coefficients when there is complex roots of the characteristic equation.
In this video we discuss how we can find a second solution to a second order equation when we already know one solution. We do this in the specific case of having one root of the characteristic equation.
In this video we learn how to solve higher order linear constant coefficient equations when the roots of the characteristic equation are distinct real roots.
In this video we solve our first nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is a polynomial.
In this video we consider the method of undetermined coefficients in general. We explain when it works and how to come up with a particular solution based off the right hand side of the differential equation (the nonhomogenous term) and the complementary solution.
In this video we consider another nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is a sum of two nonhomogenous terms. For each term we find particular solutions which then need to be added together to find the general particular solution. This video also shows what to do when there is duplication between the particular solution and the complementary solution.
In this video we use Laplace transforms to solve our first initial value problem (differential equation with initial conditions).
In this video we use the first translation theorem to find an inverse Laplace transform.
Completing the square is an important concept here.
In this video we learn how to rewrite piecewise continuous functions in terms of the unit step function.
In this video we take the Laplace transform of a piecewise continuous step function.
In this video we solve an IVP with a series of delta function terms.
In this video we define the convolution of two functions and show the convolution theorem.
We also use it to find the inverse Laplace transform of a function.
In this video we use the convolution theorem to find a nice closed integral formula for the solution of a differential equation.
In this video we use the convolution theorem to find a nice closed integral formula for the solution of a differential equation. In particular we discuss the response of a system to a delta function input. Then we use the convolution theorem to find a nice closed integral formula for the solution of general second order linear differential equation. This formula has many nice applications and can be derived very easily.
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In this video we review quickly how to solve linear second order equations with constant coefficients. We then review some information about power series.
In this video we solve our first variable coefficient differential equation using power series.
In this video we learn to solve some differential equations in Maple and also find series solutions. We plot the Airy functions as well from the last video.
In this video we solve another differential equation by finding a power series solution. We derive the recurrence relation for the coefficients. We also discuss more about initial conditions and how they determine the first two coefficients in the power series solution.We again use Maple to find the power series solutions as well. At the end we define ordinary points.
In this video we define ordinary points and singular points of second order linear differential equations. We complete one more example of finding power series solutions using the Chebyshev Differential equation. This video is continued in the next video.
Here we finish off the Chebyshev example from the previous video. We also quickly discuss Chebyshev polynomials.
We define singular and regular singular points. We begin to solve Euler's differential equation which leads to the indicial equation (we will be seeing a lot of it in this Section on series solutions.
This video continues from the previous one and we finish analyzing Euler's Differential Equation for the three different cases when considering roots of the indicial equation.
In this video we introduce the main theorem about finding Frobenius series solutions. It all revolves around the indicial equation and whether the roots differ by an integer or not. We also begin our first example of finding Frobenius series solutions when the roots differ by an integer.
We finish the example that was started in the last video. We find two guaranteed Frobenius series solutions in this example because the roots do not differ by an integer.
In this video we begin solving a differential equation where the roots of the indicial equation differ by an integer. In this case there is not always a Frobenius Series solution corresponding to the smaller root. In this example it turns out that there are two Frobenius solutions.
This video takes off from where we left off in the last video. The roots to the indicial equation differ by an integer and it turns out there are two Frobenius series solutions. We see why in this video and how to solve such examples.
In this example the roots of the indicial equation differ by an integer and there is only one Frobenius series solution corresponding to the larger root. We see why this is and learn how to solve such examples. We end the video with discussing how to find a second linearly independent solution. The answer is of course the reduction of order formula.
When the roots to the indicial equation are equal there is only one Frobenius series solution. This also may happen when the roots differ by an integer. To find a second solution we can use reduction of order. In this video (and the next) we show how to use the method of reduction formula to find y2 when passing in a series solution for y1.
We finish the reduction question from the previous video and also show how to do one more from Example 8 in a previous video.
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We begin our first example of solving the heat equation using separation of variables.
We continue the separation of variables problem from the last video. An important concept that shows up here is the linear eigenvalue problem that leads to the eigenvalues and eigenfunctions.
We continue on from the previous video. We discover the Sine Fourier series as we try to solve the heat equation.
We review the heat equation solution from the previous video and also demonstrate how to find the Sine Fourier series for a function. We plot the Fourier Series in Maple as well as the solution to the heat equation.
In this video we solve the heat equation zero flux boundary conditions. This leads to the cosine Fourier series which we also define. Separation of variables is used again to derive the eigenvalue problem.
We solve the heat equation with periodic boundary conditions. This also leads to the full general Fourier series (involving both Cosine and Sine terms). This material is continued in the next video.
A continuation of the previous video. We use orthogonality to solve for the coefficients in the Fourier series and also use Maple to animate the convergence of a Fourier Series.
We find the Fourier series for a function and also show how to use Maple to do some calculations. We also use a nice trick to show how Fourier series can be used to evaluate infinite series.
We discuss even and odd functions as well as periodic extensions (even and odd). We show how the cosine and sine Fourier series are just special cases of the more general Fourier Series.
Separation of variables is used for linear homogenous PDEs and boundary conditions. Sometimes we can do a trick to remove inhomogeneous terms and then use separation of variables. We show this for the heat equation with a non homogenous term in the PDE.
We do the same trick as the last video to remove inhomogenous terms in the boundary conditions.
In this video we solve the wave equation and look at some simulations in Maple.
We review some basics from linear algebra such as eigenvalues, eigenvectors, and symmetric matrices. We define self adjoint operators.
In this course we focus on Regular SL problems which we also define.
Learn how to solve nonhomogenous Sturm Liouville Problems.
Be sure to print off the slides so you can follow along with this section. They are added as resources to this video/lecture.
My name is Chris Levy and I have a PhD in applied mathematics from Dalhousie University. I live in Halifax, Nova Scotia, Canada.
I am a researcher, university instructor, and a budding data scientist. I have experience teaching university courses such as calculus, differential equations, and math for commerce. I have taught courses to 35 students, 70 students, and even 300 students.
I also have experience tutoring hundreds of students in mathematics. I know how to explain concepts clearly and concisely. I have been a very successful student and instructor. I know what it takes to succeed.
I enjoy hanging out with my wife and three kids, playing guitar, playing sports, and learning.