Question

1) In Lecture 12, we went into detail the proof of Picard-Lindelof Theorem. Consider the differential equation $y' = xy$ with $y(0) = 1$ a) First show that $S(x, y) = xy$ is Lipschitz on ANY rectangle in $R^2$. As a consequence, the Picard Theorem applies. Show that on the closed interval $[-\delta, +\delta]$, there exist a solution $f(x)$ and it is unique, $f(0) = 1$. b) For your solution $f(x)$, show that it is even. That is, $f(x) = f(-x)$. (Note: this must also satisfy the differential equation) c) Assume $0 < \delta < 1$, given in the interval description in part a). Let $T$ be a contraction mapping ($T$ was the mapping used to proof Picard's Theorem) acting on the continuous function $f \in C^0([-\delta, +\delta])$ $T = 1 + \int_0^x t \cdot f(t)dt$ Using this, calculate the limit with an initial constant function 1, $\lim_{n \to \infty} T^n(1)$. Show that the fixed point of $T$ is given by $exp(\frac{x^2}{2})$ d) Using the sup norm on $C^0([-\delta, +\delta])$, explain why $T^n(1)$ is Cauchy.

          1) In Lecture 12, we went into detail the proof of Picard-Lindelof Theorem.
Consider the differential equation $y' = xy$ with $y(0) = 1$
a)
First show that $S(x, y) = xy$ is Lipschitz on ANY rectangle in $R^2$. As a
consequence, the Picard Theorem applies. Show that on the closed
interval $[-\delta, +\delta]$, there exist a solution $f(x)$ and it is unique, $f(0) = 1$.
b)
For your solution $f(x)$, show that it is even. That is, $f(x) = f(-x)$.
(Note: this must also satisfy the differential equation)
c)
Assume $0 < \delta < 1$, given in the interval description in part a). Let $T$ be a
contraction mapping ($T$ was the mapping used to proof Picard's
Theorem) acting on the continuous function $f \in C^0([-\delta, +\delta])$ 
$T = 1 + \int_0^x t \cdot f(t)dt$
Using this, calculate the limit with an initial constant function
1, $\lim_{n \to \infty} T^n(1)$. Show that the fixed point of $T$ is given by $exp(\frac{x^2}{2})$
d)
Using the sup norm on $C^0([-\delta, +\delta])$, explain why $T^n(1)$ is Cauchy.

1) In Lecture 12, we went into detail the proof of Picard-Lindelof Theorem.
Consider the differential equation y' = xy with y(0) = 1
a)
First show that S(x, y) = xy is Lipschitz on ANY rectangle in R^2. As a
consequence, the Picard Theorem applies. Show that on the closed
interval [-δ, +δ], there exist a solution f(x) and it is unique, f(0) = 1.
b)
For your solution f(x), show that it is even. That is, f(x) = f(-x).
(Note: this must also satisfy the differential equation)
c)
Assume 0 < δ < 1, given in the interval description in part a). Let T be a
contraction mapping (T was the mapping used to proof Picard's
Theorem) acting on the continuous function f ∈ C^0([-δ, +δ])
T = 1 + ∫0^x t · f(t)dt
Using this, calculate the limit with an initial constant function
1, limn →∞ T^n(1). Show that the fixed point of T is given by exp((x^2)/(2))
d)
Using the sup norm on C^0([-δ, +δ]), explain why T^n(1) is Cauchy.

Added by Felipe A.

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