Chapter4, Section 3

Answers and solutions to selected problems

Problem 2

4.

5. (a)If Op(x) = f(t) is a linear DE, then Op(x) must be a linear operator and we know that any linear operator maps 0 to 0.

(b) If Op(x) is simply homogeneous then the DE Op(x) = f(t) may not be linear. For example, let Op(x) = x' + sin x , which is homogeneous since Op(0) = 0, however the DE x' + sin x = f(t) is not linear.

A first order DE of the form a(t) x' + b(t) x = f(t) is linear if a(t), b(t), and f(t) are continuous and if a(t) ~= 0 for any values of t. If this is the case then the DE may be expressed as