x0 = 0
x1 = 0.7
n = 100
y0 = 1

f[x_, y_] := (1 + y^2)*3*x

dx = (x1 - x0)/n

Y={{x0, y0}}

For[i = 0,  
      i < n,  i++,  
                       Y = 
        Append[Y,  {x0 + (i + 1)*dx, 
            Y[[i + 1, 2]] + 
              f[x0 + i*dx, Y[[i + 1, 2]]]*
                dx} ]           ]

Show[ListPlot[Y], 
      Plot[ Tan[3/2*x^2 + Pi/4],  {x, 0, 0.7}]]