# A Complete First Course in Differential Equations

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- 29.5 hours on-demand video
- 14 downloadable resources
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- Certificate of Completion

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Try Udemy for Business- Classify differential equations according to their type and order.
- Solve first order differential equations that are separable, linear, homogeneous, exact, as well as other types that can be solved through different substitutions.
- Use first order differential equations to model different applications from science.
- Solve linear second order equations with constant coefficients (both homogenous and non-homogeneous) using the method of undetermined coefficients, variation of parameters, and Laplace transforms.
- Understand the theory of linear second order differential equations and how it relates to ideas from linear algebra.
- Use linear second order equations with constant coefficients (both homogenous and non-homogeneous) to model applications from science.
- Find Laplace and inverse Laplace transforms.
- Use Laplace transforms to solve linear second order equations with constant coefficients which contain forcing functions such as impulses, step functions, and periodic functions.
- Solve systems of linear differential equations with constant coefficients and understand the importance of eigenvalues and eigenvectors for finding solutions.
- Understand the importance of the Matrix exponential and how to compute it in order to find the solutions of linear systems of differential equations.
- Apply basic numerical methods to find approximate solutions of differential equations.
- Understand the basics of some complex analysis and its usefulness to differential equations.
- Use equilibrium points, phase portraits, and stability analysis to analyze linear systems.
- Use Maple to analytically and numerically solve differential equations. Use Maple to study differential equations qualitatively.
- Model real world phenomenon with differential equations.
- Find series solutions to second order linear equations with variable coefficients. Apply this method to ordinary points and regular singular points. Find Frobenius series solutions using the method of Frobenius. Apply reduction of order to find series solutions.
- Use Fourier series to solve partial differential equations. Solve the heat, wave, and Laplace equation using separation of variables and Fourier Series. Understand theory and applications of General Fourier series, Sine Fourier series, Cosine Fourier series, and convergence of Fourier series. Solve inhomogenous PDEs.
- Use theory of vector spaces, orthogonality of functions and inner products, self adjoint operators and apply to Sturm-Liouville Eigenvalue problems. Use eigen function expansions to solve nonhomogenous problems.
- Analyze nonlinear autonomous system by finding equilibrium points and stability. Understand concept of linearization and the Hartman-Grobman Theorem. Find and analyze Hopf bifurcation as well as other commonly known bifurcations
- Apply Numerical methods and understand importance of stability and accuracy. Be able to implement in Maple. Be able to use state of the art DE solvers.

- First year differential and integral calculus

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

- First Order Differential Equations
- Linear Equations of Higher Order
- Laplace Transform Methods
- Linear Systems of Differential Equations
- Power Series Methods
- Partial Differential Equations
- Fourier Series
- Sturm Liouville Eigenvalue Problems
- Nonlinear Systems of Differential Equations
- Numerical Methods

- Students taking differential equations at college or university
- Students preparing to take differential equations at college or university
- Anyone who wants to learn about the subject of differential equations

Here we look at a differential equation which models the motion of a falling object under the force of gravity.

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 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 go through a second example of solving a first order homogeneous differential equation.

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 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 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 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 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.

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.

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.

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.

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.

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.