We study the fundamental concepts and properties associated with ODEs.

### (Video) Calculus: Single Variable

**Note: For now you can begin the video at around 1:45. We will discuss
linear and separable ODEs shortly. The midpoint method and
Runge-Kutta beginning around 12:15 are important things to understand,
but you will not be expected to compute these yourself like you will for
Euler’s method.**

### Online Texts

- OpenStax II 4.1: ODEs and Direction Fields and Numerical Methods
- Ximera OSU: ODEs and Numerical Methods

### Examples

There are three curves shown: Which of the three curves is a solution of the ODE ? In other words, which of the three curves is correctly aligned with the slope field?

couldn’t actually cross each other there.

*step size*). We use the function to dictate the slope of each piece of the approximation. Let be the solution to the initial value problem with . Use Euler’s Method with step size to approximate the value of .

- For this particular ODE, we have
- Supposing that we start at the point, , the slope will be
- Consider a line segment beginning at the point having slope which you just
calculated.
The line segment will pass through the point (use your value of to compute a numerical value of corresponding to ).

- Now we repeat: Take a new line segment beginning at the point you just
found.
Its slope is given by This new line segment passes through the point .

- Repeat again:
the line segment beginning at the point you just found will have slope and will pass through the point . Since we have arrived at an -value of , we may stop. The -value of this most recent point is our answer: