Question

Which of the following operators is linear? Justify your answer.

(a) L(y) = y' + 2ty
(b) L(y) = y″ + (1 − y²) y' + y

Answer 1

The linear operator is L(y) = y' + 2ty. It preserves addition and scalar multiplication.

Step-by-step explanation:

A linear operator is an operator that preserves addition and scalar multiplication. In other words, if L is a linear operator, then for any vectors u and v and any scalar c, we have L(u+v) = L(u) + L(v) and L(cu) = cL(u).

Let's consider the given operators:

(a) L(y) = y' + 2ty:

We can see that this operator is linear because it satisfies both properties of a linear operator. For example, let's say we have two functions u and v and a scalar c. If we compute L(u+v), we get (u+v)' + 2t(u+v) = u' + v' + 2tu + 2tv = (u' + 2tu) + (v' + 2tv) = L(u) + L(v). Similarly, if we compute L(cu), we get (cu)' + 2t(cu) = cu' + 2ctu = c(u' + 2tu) = cL(u).

(b) L(y) = y″ + (1 − y²) y' + y:

This operator is not linear because it does not satisfy the property of preserving scalar multiplication. If we compute L(cu), we get (cu)″ + (1 − (cu)²)(cu)' + cu = c(u″ + (1 − u²)u' + u) ≠ cL(u).

Therefore, the operator (a) L(y) = y' + 2ty is linear.