1. If the functions ao ao(x), a1 = a1(x) and g = g(x) are continuous on R then the IVP y" + a1(x)y' + ao(x)y = g(x), y(1) = yo, y'(1) = Y1 has the unique solution y = y(x) for every pair of real numbers yo, Y1. Moreover the interval of definition of this solution is the whole real line. TRUE FALSE 2. Every fundamental set of solutions of a homogeneous DE of order n has exactly n solutions. TRUE FALSE 3. The functions f1(x) := 1+x, f2(x) := x², f3(x) := ln(x) are linearly dependent (on the interval (0, 0)). TRUE FALSE
1. If the functions ao ao(x), a1 = a1(x) and g = g(x) are continuous on R then the IVP y" + a1(x)y' + ao(x)y = g(x), y(1) = yo, y'(1) = Y1 has the unique solution y = y(x) for every pair of real numbers yo, Y1. Moreover the interval of definition of this solution is the whole real line. TRUE FALSE 2. Every fundamental set of solutions of a homogeneous DE of order n has exactly n solutions. TRUE FALSE 3. The functions f1(x) := 1+x, f2(x) := x², f3(x) := ln(x) are linearly dependent (on the interval (0, 0)). TRUE FALSE
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please answer the following true and false questions.
![### Differential Equations and Linear Dependence: A Quiz
**1. If the functions \( a_0 = a_0(x) \), \( a_1 = a_1(x) \), and \( g = g(x) \) are continuous on \( \mathbb{R} \), then the IVP**
\[ y'' + a_1(x)y' + a_0(x)y = g(x), \quad y(1) = y_0, \quad y'(1) = y_1 \]
**has the unique solution \( y = y(x) \) for every pair of real numbers \( y_0, y_1 \). Moreover, the interval of definition of this solution is the whole real line.**
TRUE FALSE
<br>
**2. Every fundamental set of solutions of a homogeneous DE of order \( n \) has exactly \( n \) solutions.**
TRUE FALSE
<br>
**3. The functions**
\[ f_1(x) := 1 + x, \quad f_2(x) := x^2, \quad f_3(x) := \ln(x) \]
**are linearly dependent (on the interval \( (0, \infty) \)).**
TRUE FALSE
<br>
**4. If \( \mathcal{L}(f) \) exists then \( \mathcal{L}(e^{at}f(t)) \) also exists for every real number \( a \).**
TRUE FALSE
<br>
**5. If a function \( f = f(t) \) is continuous on \( [0, \infty) \) and is of exponential order then \( \mathcal{L}(f) \) exists.**
TRUE FALSE
<br>
**6. If \( \mathcal{L}(f) = \mathcal{L}(g) \) then \( f(t) = g(t) \) for every \( t \geq 0 \).**
TRUE FALSE](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa061449e-1e75-4fa5-9e02-82a00bd019ab%2Fe39dd085-c066-4a1c-ae9a-97c0d537b3ad%2F0iy2diq.png&w=3840&q=75)
Transcribed Image Text:### Differential Equations and Linear Dependence: A Quiz
**1. If the functions \( a_0 = a_0(x) \), \( a_1 = a_1(x) \), and \( g = g(x) \) are continuous on \( \mathbb{R} \), then the IVP**
\[ y'' + a_1(x)y' + a_0(x)y = g(x), \quad y(1) = y_0, \quad y'(1) = y_1 \]
**has the unique solution \( y = y(x) \) for every pair of real numbers \( y_0, y_1 \). Moreover, the interval of definition of this solution is the whole real line.**
TRUE FALSE
<br>
**2. Every fundamental set of solutions of a homogeneous DE of order \( n \) has exactly \( n \) solutions.**
TRUE FALSE
<br>
**3. The functions**
\[ f_1(x) := 1 + x, \quad f_2(x) := x^2, \quad f_3(x) := \ln(x) \]
**are linearly dependent (on the interval \( (0, \infty) \)).**
TRUE FALSE
<br>
**4. If \( \mathcal{L}(f) \) exists then \( \mathcal{L}(e^{at}f(t)) \) also exists for every real number \( a \).**
TRUE FALSE
<br>
**5. If a function \( f = f(t) \) is continuous on \( [0, \infty) \) and is of exponential order then \( \mathcal{L}(f) \) exists.**
TRUE FALSE
<br>
**6. If \( \mathcal{L}(f) = \mathcal{L}(g) \) then \( f(t) = g(t) \) for every \( t \geq 0 \).**
TRUE FALSE
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
