1. Section 17, Problem 5(b)
Important! We will revisit this type of non-autonomous ODE, called Euler’s equidimensional equation, in the PDE section. Also, use the textbook hint to solve this.
2. Write the following linear di↵erential equations with constant coefficients in the form of a system of first order linear di↵erential equations and solve:
(a) x¨ + x˙ − 2x = 0; x(0) = x0, x˙(0) = v0
(b) x¨ + x = 0; x(0) = x0, x˙(0) = v0
Compare the solution of x(t) to the one obtained using the method in Section 17.
3. Show that the initial value problem
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x˙ = |x| 2 , x(0) = 0
has four di↵erent solutions through the point (0, 0). Sketch these solutions in the (t, x)-plane.
Carefully check the domain of the solutions obtained.
4. Section 69, Problem 3. Plot the solutions. Irrespective of the chosen initial approximation, the Picard iterates seem to converge, why?
Hint: For (c), approximate cos x by taking appropriate number of terms of its Taylor series.
5. Section 71, Problem 1
6. (Following problem will not be graded, solving will help undestand proof of Picard’s theorem)
a. Give an example of a converging sequence of continuous functions whose limit is not continuous.
b. State a condition that assures that the limiting function is continuous.
c. Give a mathematical definition of the key term(s) used in b.
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