Worksheet - Differential Equations

Worksheet - Differential Equations

• 1.
  Worksheet – DifferentialEquations 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.