Math 6410 • ODE

The goal of this problem is to use Picard iterates to find the solution of the initial
value problem
\[
  \frac{dx}{dt}=x^2,\qquad x(0)=1
  \tag{1}
\]
\begin{enumerate}
\item[(a)] State a theorem, without proof, with necessary conditions on \(f(x,t)\),
guaranteeing the existence of a unique solution of the differential equation
\(\frac{dx}{dt}=f(x,t)\), \(x(0)=x_0\).
\item[(b)] Verify that the function \(f=x^2\) satisfies these necessary conditions.
\item[(c)] Define the Picard iterates for the equation (1) and calculate the first
three functions \(x_0(t)\), \(x_1(t)\), and \(x_2(t)\).
\item[(d)] Prove by induction that
\[
  x_n(t)=1+t+t^2+\cdots+t^n+O(t^{n+1})
  =\sum_{j=0}^{n}t^j+O(t^{n+1}).
  \tag{2}
\]
Hint: You will need to show that
\[
  \sum_{j=0}^{n}t^j\sum_{k=0}^{n}t^k
  =\sum_{j=0}^{n}jt^j+O(t^{n+1}).
\]
\item[(e)] Find the exact solution of (1). How does it compare to the approximate
solution found in the previous part?
\end{enumerate}