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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
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Section 1: Introduction to Differential Equations and their Applications  

Lecture 1  06:55  
Here we look at a differential equation which models the motion of a falling object under the force of gravity. 

Lecture 2  04:11  
Here we look at a differential equation which models the motion of a falling object under the force of gravity and air resistance. 

Lecture 3  07:03  
Here we look at a differential equation which models the motion of mass on a spring. 

Lecture 4  07:23  
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. 

Lecture 5  06:45  
Here we look at a differential equation which models the motion of a simple pendulum. 

Lecture 6  12:45  
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. 

Lecture 7  13:30  
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. 

Lecture 8  07:42  
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. 

Lecture 9  04:18  
Here we show that solutions to differential equations can be explicit or implicit. 

Section 2: First Order Differential Equations  
Lecture 10  11:28  
In this video we learn about slope fields and solution curves for first order differential equations. 

Lecture 11  10:45  
In this video we learn about the existence and uniqueness theorem for first order differential equations. 

Lecture 12  06:14  
In this video we define what separable differential equations are and learn how to solve them. 

Lecture 13  05:39  
In this video we solve a separable differential equation. We also quickly review integration by parts. 

Lecture 14  07:05  
In this video we discuss the differential equation for Newtons law of cooling and solve it. We can solve the equation because its separable. 

Lecture 15  05:57  
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. 

Lecture 16  10:16  
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. 

Lecture 17  13:02  
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. 

Lecture 18  14:52  
In this video we learn to solve linear first order differential equations. 

Lecture 19  07:47  
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. 

Lecture 20  09:17  
In this video we do an example of a mixing problem with one tank. 

Lecture 21  07:22  
In this video we learn how to solve exact differential equations. 

Lecture 22  10:23  
In this video we go through an example of solving an exact differential equation. 

Lecture 23  11:54  
In this video we go through another example of solving an exact differential equation. 

Lecture 24  06:48  
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. 

Lecture 25  09:09  
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. 

Lecture 26  08:58  
In this video we go through an example of solving a first order homogeneous differential equation. 

Lecture 27  03:26  
In this video we go through a second example of solving a first order homogeneous differential equation. 

Lecture 28  08:04  
In this video we learn how to solve Bernoulli Differential Equations. They can be solved by an appropriate substitution. 

Lecture 29  10:56  
In this video we see how substitutions can sometimes reduce second order differential equations to first order differential equations which we can then solve. 

Lecture 30  14 pages  
This assignment tests your understanding of the material presented in Sections 1 and 2. 

Section 3: Higher Order Differential Equations  
Lecture 31  05:41  
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 nonhomogeneous equations. 

Lecture 32  04:28  
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. 

Lecture 33  07:01  
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. 

Lecture 34  04:17  
In this video we state the existence and uniqueness theorem for higher order differential equations. 

Lecture 35  11:44  
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. 

Lecture 36  09:20  
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. 

Lecture 37  02:01  
In this video we summarize the main points of interest regarding the solutions of linear second order homogeneous equations. 

Lecture 38  12:02  
Here we introduce the idea of linear independence and dependence for two functions and relate it to the Wronskian of two functions. 

Lecture 39  05:32  
In this video we look at a stronger theorem relating linear independence/dependence to the wronskian for solutions of second order linear homogenous equations. 

Lecture 40  07:48  
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. 

Lecture 41  03:56  
In this video we begin to learn how to solve second order homogeneous equations with constant coefficients. 

Lecture 42  04:26  
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. 

Lecture 43  04:49  
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. 

Lecture 44  11:30  
In this video we learn how to solve second order homogeneous equations with constant coefficients when there is complex roots of the characteristic equation. 

Lecture 45  12:30  
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. 

Lecture 46  03:24  
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. 

Lecture 47  03:57  
In this video we learn how to solve higher order linear constant coefficient equations when the some of the roots of the characteristic equation are repeated real roots.  
Lecture 48  02:13  
In this video we learn how to solve higher order linear constant coefficient equations when the roots of the characteristic equation are complex (but not repeated).  
Lecture 49  03:37  
In this video we learn how to solve higher order linear constant coefficient equations when the roots of the characteristic equation are complex and repeated.  
Lecture 50  02:57  
In this video we look at solving one more higher order homogeneous equation with constant coefficients. We see how the different cases from the last few videos can be used together in one differential equation.  
Lecture 51  03:18  
In this video we begin to discuss the solutions of nonhomogenous differential equations.  
Lecture 52  08:48  
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. 

Lecture 53  03:48  
In this video we solve another nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is an exponential function.  
Lecture 54  07:05  
In this video we solve another nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is a cosine function.  
Lecture 55  08:32  
In this video we consider another nonhomogenous differential equation and try using the method of undetermined coefficients as we have done in the last few videos. In turns out that we have to modify our guess of the particular solution when terms in it also appear in the complementary solution.  
Lecture 56  03:32  
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. 

Lecture 57  07:50  
In this video we solve another nonhomogenous differential equation using the method of undetermined coefficients. We do this when the right hand side of the differential equation is an exponential function multiplied by a cosine function.  
Lecture 58  06:23  
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.  
Lecture 59  04:10  
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. 

Lecture 60 
Assignment 2

4 pages  
Lecture 61  09:45  
If solving a linear second order homogenous differential equation, we can always find a second solution if we already know one solution. This is done through the method of reduction of order. We had considered this method for a specific example in an earlier video. Here we derive a formula in the general case which can be used to find the second solution.  
Lecture 62  04:11  
In this video we use the reduction of order formula to find a second solution for a specific example.  
Lecture 63  15:32  
In this video we derive a formula for finding a particular solution to a nonhomogenous differential equation. This method is an alternative method to the method of undetermined coefficients.  
Lecture 64  05:20  
In this video we use the method of variation of parameters to solve a nonhomogenous differential equation.  
Lecture 65 
Assignment 3

4 pages  
Section 4: Laplace Transforms  
Lecture 66  05:17  
In this video we define the Laplace transform and discuss the types of differential equations we will be solving with it.  
Lecture 67 
Laplace Transform Example: Unit Step Function

04:50  
Lecture 68 
Laplace Transform Example: First Derivative

04:27  
Lecture 69  03:10  
In this video we find the Laplace transform of the second derivative of a function.  
Lecture 70 
Existence of the Laplace Transform

08:08  
Lecture 71  02:27  
In this video we find the Laplace transform of the exponential function.  
Lecture 72 
Laplace Transform Example: Cosine, Sine, Hyperbolic Cosine and Sine

08:18  
Lecture 73  03:58  
In this video we define the inverse Laplace transform.  
Lecture 74  11:10  
In this video we use Laplace transforms to solve our first initial value problem (differential equation with initial conditions). 

Lecture 75  08:30  
In this video we use Laplace transforms to solve our second initial value problem (differential equation with initial conditions).  
Lecture 76  12:35  
In this video we look at some more techniques to invert Laplace transforms. We use partial fractions along with a table of Laplace transforms.  
Lecture 77  05:46  
Here we learn a very important theorem, the First Translation Theorem. It is useful in finding inverse Laplace transforms that are translated/shifted.  
Lecture 78  08:35  
In this video we use the first translation theorem to find an inverse Laplace transform.  
Lecture 79  07:59  
In this video we use the first translation theorem to find an inverse Laplace transform. Completing the square is an important concept here. 

Lecture 80  03:23  
In this video we learn about another very important translation theorem which we will call the Second Translation Theorem. It involves finding the transforms of unit step functions which are multiplied by translated functions.  
Lecture 81  06:59  
In this video we learn how to rewrite piecewise continuous functions in terms of the unit step function. 

Lecture 82  04:16  
In this video we take the Laplace transform of a piecewise continuous step function. 

Lecture 83  05:29  
In this video we take the Laplace transform of a piecewise continuous step function.  
Lecture 84  19:45  
In this video we solve a complicated IVP with a non homogenous term that is piecewise continuous.  
Lecture 85  13:12  
In this video we solve a complicated IVP with a non homogenous term that is piecewise continuous.  
Lecture 86 
Assignment 4

5 pages  
Lecture 87  07:42  
In this video we look at a formula for the derivative of transforms and one application of it.  
Lecture 88  05:30  
In this video we find out how to take the Laplace Transform of periodic piecewise functions.  
Lecture 89  16:42  
In this video we solve an IVP with a periodic piecewise nonhomogenous term.  
Lecture 90  10:55  
In this video we define the Dirac Delta Function which can be used to model impulses.  
Lecture 91  09:10  
In this video we solve an IVP with a delta function term.  
Lecture 92  05:19  
In this video we solve an IVP with a series of delta function terms. 

Lecture 93  06:58  
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. 

Lecture 94  04:18  
In this video we use the convolution theorem to find a nice closed integral formula for the solution of a differential equation. 

Lecture 95  06:31  
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. 

Lecture 96 
Assignment 5

7 pages 
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.