University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2021
Views:

413

Pages:

1
Author:

Annabelle Moreno

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.

Extra Practice Problems Linear Operators and Cramer’s Rule

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