In mathematics, a differential equation refers to an equation that relates one or more unknown functions and their derivatives. In practice, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Differential equations play a prominent role in many disciplines of science and engineering.

The study of differential equations consists of the study of their solutions, and of the properties of their solutions. This course aims to provide certain methods for finding solutions for some linear equations of smaller orders. It starts with a quick review of integral calculus to remind the reader of some well-known and useful results in that subject to be used later on. Then we continue with the following topics:

1. First-order Differential Equations.

In this section, we first study separable equations, and then the exact equations. Here the integrating factors play a crucial role in finding the solutions.

2. Linear Differential Equations.

In this section, we introduce the notion of a fundamental set and give a theorem saying that a fundamental set exists for any linear differential equation. We also give a formula for the second solution of a linear equation if we have already known one solution of the equation.

3. Second-order DEs with Constant Coefficients

In this section, we study the second-order differential equations with constant coefficients. For any such equation, we can always find solutions. We will also learn the Cauchy-Euler equations because such equations have similar results to the first case.

4. Power Series Solutions

It is shown that at an ordinary point, we can always find two linearly independent power series solutions for a linear equation of order 2. For the solutions at singular points, we use the Frobenius Theorem. We will also study Bessel's equations and Legendre's equations.

5. The Laplace Transform

In this section, we introduce the Laplace Transform and the Inverse Transform and show that how to use them to solve differential equations. We will also study the general solutions of non-homogeneous differential equations.

6. Systems of differential equations

In this section, we will show how to use results from Matrix Theory to solve Systems of differential equations.

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