Explanation Approach: To show the differential operator, a linear operator, verify addition properties and scalar multiplication properties. Calculation: Let’s consider the differential operator, L [ y ] : = d n y d x n + p 1 d n − 1 y d x n − 1 + ... + p n y defined on the set of all n -times differentiable function on interval ( a , b ) Now to become the differential operator linear it should verify both addition and scalar multiplication properties 1. Addition properties Suppose y 1 , y 2 , ...... , y m are n -times differentiable functions on interval ( a , b ) Consider, L [ y 1 + y 2 + ...... + y m ] = ( d n d x n + p 1 d n − 1 d x n − 1 + ....... + p n ) [ y 1 + y 2 + ...... + y m ] Denote d n d x n by D n Then L [ y 1 + y 2 + ...... + y m ] = ( D n + p 1 D n − 1 + ...... + p n ) [ y 1 + y 2 + ...... + y m ] = ( D n + p D n − 1 + ...... + p n ) [ y 1 ] + ..... + ( D n + p 1 D n − 1 +

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 6.1, Problem 25E

To determine

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The differential operator L is a linear operator by verifying addition and scalar multiplication properties.

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Chapter 6.1, Problem 24E

Chapter 6.1, Problem 26E

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Fundamentals of Differential Equations and Boundary Value Problems

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To prove/verify: The differential operator L is a linear operator by verifying addition and scalar multiplication properties.

Solution Summary: The author proves that the differential operator L is a linear operator by verifying addition and scalar multiplication properties.