Picard Approximation Consider the initial value problem 
$$\frac{dy}{dx} = y^2 + 3x^2 - 1, \\
y(1) = 1$$
on D = {|x-1| <= 1, |y-1| <= 1}
Find the second approximation to the solution and estimate the error term
If someone could maybe walk me through the steps that would be most appreciated =)

We are given:
$$\tag 1 \frac{dy}{dx} = y^2 + 3x^2 - 1, y(1) = 1, ~\mbox{on}~~ D = {|x-1| \le 1, |y-1| \le 1}$$
The Picard-Lindelöf iteration is given by:
$$\tag 2 \displaystyle y_0(x) = y_0, ~~y_{n+1}(x) = y_0 + \int^x_{x_0} f(s, y_n(s))ds$$
For $(1)$, we have: $f(s, y_n(s)) = y_n^2+3s^2-1$ and using $(2)$, yields:


*

*$\displaystyle y_0(1) = y_0 = 1$

*$\displaystyle y_1(x) = y_0 + \int^x_{x_0} f(s, y_0(s))ds = 1 + \int^x_{1} ((1)^2+3s^2-1) ds = x^3$

*$\displaystyle y_2(x) = y_0 + \int^x_{x_0} f(s, y_1(s))ds = 1 + \int^x_{1} ((s^3)^2+3s^2-1) ds = \dfrac{x^7}{7}+x^3-x +\dfrac{6}{7}$
If you need one more iteration, you would get:
$$y_3(x) = \frac{x^{15}}{735}+\frac{2 x^{11}}{77}-\frac{2 x^9}{63}+\frac{3 x^8}{98}+\frac{x^7}{7}-\frac{2 x^5}{5}+\frac{3 x^4}{7}+\frac{4 x^3}{3}-\frac{6 x^2}{7}-\frac{13 x}{49}+\frac{4099}{6930}$$
