b. Verify that the given function is a solution of the differential equation. z(t) = e-t + e2t + e-3t – et z" (t) + 2 z"(t) – 57' (t) – 67(t) = 8et
b. Verify that the given function is a solution of the differential equation. z(t) = e-t + e2t + e-3t – et z" (t) + 2 z"(t) – 57' (t) – 67(t) = 8et
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...
Related questions
Question
![### Verification of a Function as a Solution to a Differential Equation
**Problem Statement:**
Verify that the given function is a solution of the differential equation.
**Function:**
\[ z(t) = e^{-t} + e^{2t} + e^{-3t} - e^t \]
**Differential Equation:**
\[ z'''(t) + 2z''(t) - 5z'(t) - 6z(t) = 8e^t \]
**Goal:**
To determine if \( z(t) \) satisfies the differential equation, we need to compute the derivatives of \( z(t) \), substitute them into the differential equation, and verify that the left-hand side equals the right-hand side, \( 8e^t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c9a3da8-b443-4c7b-9ddb-d4ba4f71dd14%2F72bff2d7-4f64-4bee-a4fa-9eaefd8b324a%2Fvq8y2i9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Verification of a Function as a Solution to a Differential Equation
**Problem Statement:**
Verify that the given function is a solution of the differential equation.
**Function:**
\[ z(t) = e^{-t} + e^{2t} + e^{-3t} - e^t \]
**Differential Equation:**
\[ z'''(t) + 2z''(t) - 5z'(t) - 6z(t) = 8e^t \]
**Goal:**
To determine if \( z(t) \) satisfies the differential equation, we need to compute the derivatives of \( z(t) \), substitute them into the differential equation, and verify that the left-hand side equals the right-hand side, \( 8e^t \).
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning