In this lesson, we will look at the notation and highest order of differential equations. To find the highest order, all we look for is the function with the most derivatives. After, we will verify if the given solutions is an actual solution to the differential equations. We do this by simply using the solution to check if the left hand side of the equation is equal to the right hand side. Lastly, we will look at an advanced question which involves finding the solution of the differential equation.

## Differential Equations

**1. **__Introduction to Differential equations__

__Introduction to Differential equations__

**a. Order and solutions to differential equations**

This lesson is an introduction to concepts we will be using throughout the whole Differential Equations course.

**2. **__First Order Differential Equations __

__First Order Differential Equations__

**a. Slope fields**

**b. Separable Equations**

**c. Exact differential equations**

**d. Integrating factor technique**

**e. Bernoulli equations**

**f. Interval of validity**

**g. Modeling with differential equations**

**h. Equilibrium solutions**

**i. Eulers method**

**3. **__Second Order Differential Equations __

__Second Order Differential Equations__

**a. Homogeneous linear second order differential equations**

**b. Characteristic equation with real distinct roots**

A similar process will be used for the next two lessons, but the special case for today is that we will find real distinct roots.

**c. Characteristic equation with complex roots**

**d. Characteristic equation with repeated roots**

**e. Reduction of order**

**f. Wronskian**

**g. Method of undetermined coefficients**

**h. Applications of second order differential equations**

**4. **__Laplace Transforms __

__Laplace Transforms__

**a. Introduction to the Laplace transform**

**b. Calculating Laplace transforms**

**c. Inverse Laplace transforms**

**d. Solving differential equations with the Laplace transform**

**e. Step functions**

**f. Solving differential equations with step functions**

**g. Dirac Delta function**

**h. Convolution integral**

We say that:

$y' (x) = \frac{dy}{dx}$ or $y'(t) = \frac{dy}{dt}$

Where:

1. $y'(x)$ is the first derivative of the function y in terms of $x$.

2. $y'(t)$ is the first derivative of the function y in terms of $t$.

$y' (x) = \frac{dy}{dx}$ or $y'(t) = \frac{dy}{dt}$

Where:

1. $y'(x)$ is the first derivative of the function y in terms of $x$.

2. $y'(t)$ is the first derivative of the function y in terms of $t$.