A Complete First Course in Differential Equations

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.