Explanation to Euler's Method

Question

I am trying to figure out Euler's method.

The initial value problem is : $$P'(t)=0.7P(t)(1-\frac{P(t)}{750})-20, P(0)=30$$

The time step is set to $Δt=7$ days

For the algorithm we have:

$f(t,P)=0.7P(1-\frac{P}{750})-20$

$h=7$

$t_0=0$

$w_0=P(t_0)=P(0)=30$.

Could someone explain the next step? I am confused

Answer 1

Euler's method reads in this case $$ P_0 = 30, \quad P_{i+1} = P_i + h \left( 0.7 P_i \left(1- \frac{P_i}{750}\right) -20\right) $$

So, in your example, $$ P(7)\approx P_0 + 7 \left(0.7 P_0 \left(1- \frac{P_0}{750}\right) -20\right) $$