We use Euler’s method to find an approximate solution to

with initial value

First, we set

deltat = 0.05;

(The semi colon just suppresses the output.)

Next we define the function

f[t_, P_] := (1 + t) P^2;

Now we set up the recursion relation. We want the initial values to be

and

and the rest of the values to be given by

and

The code which makes a table of such values is this

values = RecurrenceTable[{ t[k + 1] == t[k] + deltat, P[k + 1] == P[k] + deltat*f[t[k], P[k]], t[0] == 0, P[0] == 3 }, {t, P}, {k, 0, 10}]

Here we only have the first time steps.

To make a list of values, we use the code

Grid[values]

To make a plot, use the code

ListLinePlot[values, PlotMarkers -> Automatic]