This document contains a worksheet with problems involving differential equations. It asks the student to: 1) Compute the general solutions to three differential equations. 2) Apply Picard-Lindelöf theorem to verify existence and uniqueness of solutions for an initial value problem, and compute Picard iterates to approximate the solution. 3) Analyze an equation involving trigonometric functions by studying equilibrium points and stability. 4) Prove that autonomous differential equations cannot have periodic solutions.

1. Worksheet – Differential Equations
1. Compute the general solution of each of the following differential equations, where x = x(t).
1.1. x = xt
1.2. x + x = t2
1.3. x + x = t
2. Consider the following initial value problem:
x = x2
x(1) = 1
and the rectange R = (t, x) ∈ R2
: |t − 1| ≤ a ∧ |x − 1| ≤ b .
2.1. Verify that the assumptions of Picard–Lindel¨of theorem are satisfied for the above initial value
problem so that there is a unique solution whose graph lies in R.
2.2. Show that Picard–Lindel¨of theorem guarantees the existence of a solution for the IVP that is defined
on an interval whose amplitude does not exceed 1/2.
2.3. Take a = b = 4. Find the interval I of existence of solution based on Picard–Lindel¨of theorem.
Compute the Picard iterate of second order (i.e., x2(t)) and find a majorant for the error |x(t)−x2(t)|,
for each t ∈ I.
2.4. Solve the inital value problem and comment the above results.
3. Consider the differential equation x = cos x + sin x.
3.1. Show that each solution x(t) of the differential equation is Lipschitz continuous with respect to t ∈ I,
where I denotes its maximal interval of existence.
3.2. Find the equilibrium points for the above equation and study their stability.
4. Show that an autonomous differential equation cannot have periodic solutions.