5.1 Definition

An ordinary differential equation (ODE) contains one or more derivatives of an unknown function of one variable.

The order of an ODE is its highest derivative. First-order ODEs in implicit form look like:

and in explicit form look like:

When do we see first-order ODEs in chemical engineering?

• Unsteady mass/energy balances

• Chemical kinetics

Direction fields¶

ODEs have families of solutions due to integration constants. The direction field (or slope field) is obtained by plotting the slope at (x, y) from $y' = f(x, y)$, and it shows how a point will evolve.

Example: Direction field¶

Given the first-order ODE

Draw the slope field, then verify that

is a general solution.

The direction field is

Now, let’s verify the proposed solution. The first derivative of y is

Substituting y and $y'$ in the differential equation

shows this is indeed a solution.

Initial value problems¶

To obtain a specific solution, an initial condition specifying a point $y(x_0) = y_0$ is needed. A first-order ODE with an initial condition is called an initial value problem.

Example: Initial value problem¶

For the first-order ODE above, find the solution when $y(0) = 1$.

Substitute $x = 0$ and $y = 1$ into the general solution $y = c e^{-x^2/2}$ to determine $c$:

This gives the solution