3. For any real number $ a $, verify that the functions $ y_1 = a + \cos x $ and $ y_2 = a + \sin x $ are solutions of the following differential equation $ y'' + y = a $. Consider $ c_1 $ and $ c_2 $ are two constants. Answer the following.(a) Is $ c_1y_1 + c_2y_2 $ a solution for $ a = 1 $ and $ c_1 + c_2 \neq 1 $? (justify your answer). No points will be given for the answers without justifications.(b) Is $ c_1y_1 + c_2y_2 $ a solution for $ a = 0 $? (justify your answer). No points will be given for the answers without justifications.(c) For $ a = 0 $, solve the differential equation with initial condition $ y(0) = 2 $ and $ y'(0) = 3 $.

First let's check if y1 and y2 are solution of given differential equation by substituting y1 and y2…

Answered: 3. For any real number a, verify that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. For any real number $ a $, verify that the functions $ y_1 = a + \cos x $ and $ y_2 = a + \sin x $ are solutions of the following differential equation $ y'' + y = a $. Consider $ c_1 $ and $ c_2 $ are two constants. Answer the following.

(a) Is $ c_1y_1 + c_2y_2 $ a solution for $ a = 1 $ and $ c_1 + c_2 \neq 1 $? (justify your answer). No points will be given for the answers without justifications.

(b) Is $ c_1y_1 + c_2y_2 $ a solution for $ a = 0 $? (justify your answer). No points will be given for the answers without justifications.

(c) For $ a = 0 $, solve the differential equation with initial condition $ y(0) = 2 $ and $ y'(0) = 3 $.

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First let's check if y₁and y₂are solution of given differential equation by substituting y₁and y₂in left side of differential equation and checking if it's equals to right hand side.

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