Extra Practice Problems
Linear Operators and Cramer’s Rule
1. Let R(f ) be the operator defined by: R(f ) = f 00 (t) + 3t2 f (t). Find R(f ) for each
function below:
(a) f (t) = t2
(b) f (t) = sin(3t)
(c) f (t) = 2t − 5
2. Let R be the operator defined in the previous problem. Show that R is a linear operator.
3. Let F (y) = y 00 + y − 5. Explain why F is not linear.
4. Find the operator associated with the given differential equation, and classify it as
linear or not linear:
(a) y 0 = ty 2 = cos(t)
(b) y 00 = 4y 0 + 3y + sin(t)
(c) y 0 = et y + 5
(d) y 00 = − cos(y) + cos(t)
5. Use Cramer’s Rule to solve the following systems:
(a)
C1 + C2 = 2
−2C1 − 3C2 = 3
(b)
C1 + C2 = y 0
r1 C 1 + r2 C 2 = v 0
(c)
C1 + C2 = 2
3C1 + C2 = 1
(d)
2C1 − 5C2 = 3
6C1 − 15C2 = 10
(e)
2x − 3y = 1
3x − 2y = 1
6. Suppose L is a linear operator. Let y1 , y2 each solve the equation L(y) = 0 (so that
L(y1 ) = 0 and L(y2 ) = 0). Show that anything of the form c1 y1 + c2 y2 will also solve
L(y) = 0.
Go back to Section 2.4, page 76 and look at Exercises 23-26. This section generalizes
those results.