Please show all work and do all parts. Only do question 6 3 Second and Higher Order Linear Differential EquationsExercises 3-12:For the given differential equation,(a) Determine the roots of the characteristic equation.(b) Obtain the general solution as a linear combination of real-valued solutions.(c) Impose the initial conditions and solve the initial value problem.3. y" + 4y = 0, y(π/4)= -2,y'(π/4) = 14. y" + 2y + 2y = 0, y(0) = 3,y'(0) = -15. 9y"+y = 0, y(π/2) = 4, y' (π/2) = 06. 2y" - 2y + y = 0, y(-л) = 1, y'(-) = -17. y" + y + y = 0,= 0,8. y" + 4y + 5y = 0,9. 9y" + 6y + 2y = 0,10. y" +4n²y = 0, y(1) = 2, y'(1) = 111. y" - 2√2y' + 3y = 0, y(0) = -1/2, y'(0) = √212. 9y" +²y = 0, y(3) = 2, y'(3) = -Exercises 13-21:y(0) = -2₁ (0) = -2₁ offy(π/2) = 1/2, y'(π/2) = -2y(3л) = 0, y'(3л) = 1/3

Answered: 6. 2y" - 2y + y = 0, y(-) = 1, y'(-) =…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Please show all work and do all parts. Only do question 6

3 Second and Higher Order Linear Differential Equations
Exercises 3-12:
For the given differential equation,
(a) Determine the roots of the characteristic equation.
(b) Obtain the general solution as a linear combination of real-valued solutions.
(c) Impose the initial conditions and solve the initial value problem.
3. y" + 4y = 0, y(π/4)= -2,
y'(π/4) = 1
4. y" + 2y + 2y = 0, y(0) = 3,
y'(0) = -1
5. 9y"+y = 0, y(π/2) = 4, y' (π/2) = 0
6. 2y" - 2y + y = 0, y(-л) = 1, y'(-) = -1
7. y" + y + y = 0,
= 0,
8. y" + 4y + 5y = 0,
9. 9y" + 6y + 2y = 0,
10. y" +4n²y = 0, y(1) = 2, y'(1) = 1
11. y" - 2√2y' + 3y = 0, y(0) = -1/2, y'(0) = √2
12. 9y" +²y = 0, y(3) = 2, y'(3) = -
Exercises 13-21:
y(0) = -2₁ (0) = -2₁ off
y(π/2) = 1/2, y'(π/2) = -2
y(3л) = 0, y'(3л) = 1/3

Expert Solution

Step 1

Introduction:

The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients.

A homogeneous differential equation is an equation containing a differentiation and a function, with a set of variables. The function f(x, y) in a homogeneous equation is zero.

Given: $2 y'' - 2 y' + y = 0$ , $y (- π) = 1, y' (- π) = - 1$