Determing if Euler's Method Overestimates

Question

How do you work out if Euler's method overestimates the actual solution, for the ODE:

$\frac{dy}{dx}=24\tan(\pi x)$

With steps of 0.25 from $1\le x\le 2$?

Answer 1

$dy = 24\tan({\pi x})dx \Leftrightarrow y(x) = \int 24\tan({\pi x})dx \Leftrightarrow \frac{24}{\pi}\int \tan(u)du = \frac{24}{\pi}\int \frac{sin(u)}{cos(u)} $

Let $s = \cos(u)$ then $ds = -\sin(u)du$ so you get :

$ -\frac{24}{\pi}\int \frac{1}{s}ds = -\frac{24}{\pi}\ln(s) + c $

Substitute $s$ back in and you get :

$ -\frac{24}{\pi}\ln(s) + c = -\frac{24}{\pi}\ln(cos(πx)) + c$

Now you can evaluate anything you want.