]> Exercises

# Exercises

1. What is the solution of this ODE: $\frac{dy\left(t\right)}{dt}=\lambda y\left(t\right)$ ?

$y\left(t\right)={y}_{0}\mathrm{exp}\left(\lambda t\right)$

2. How are the solutions of these two ODEs different?
$\frac{dy\left(t\right)}{dt}=\lambda y\left(t\right)$ and $\frac{df\left(x\right)}{dx}=\beta f\left(x\right)$

The solutions are the SAME. The names of the unknown functions and variables have been changed but this does not fundamentally change the solution. The same common ODEs can be found in very different branches of engineering. If you know the solution in one case, then you also know it in the other cases. Your physical interpretation of the solution will be different, but the mathematics is the same.

3. If you can’t find the analytic function that is the solution of an ODE, then what else can you do?

Find a NUMERICAL solution to the ODE.

4. What makes an ODE first order?

The ODE involves terms that include relationships between the unknown function and its FIRST derivative.