y" +9y = 2x²e³x +5

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...
icon
Related questions
Question
Solve the following differential equation.
**Differential Equation Analysis**

The equation displayed is:

\[ y'' + 9y = 2x^2 e^{3x} + 5 \]

### Explanation:

This is a second-order non-homogeneous linear differential equation, where:

- \( y'' \) represents the second derivative of \( y \) with respect to \( x \).
- \( 9y \) is a term where the function \( y \) is multiplied by 9.
- The right-hand side, \( 2x^2 e^{3x} + 5 \), is the non-homogeneous part of the equation. 
  - \( 2x^2 e^{3x} \) represents a term with a polynomial of degree 2 in \( x \) multiplied by an exponential function.
  - The constant 5 is an external forcing term.

This equation might be analyzed or solved to find \( y(x) \), the function describing a particular physical or theoretical system influenced by the given non-homogeneity. Solutions typically involve finding both the complementary solution (related to the homogeneous equation) and a particular solution for the non-homogeneous part.
Transcribed Image Text:**Differential Equation Analysis** The equation displayed is: \[ y'' + 9y = 2x^2 e^{3x} + 5 \] ### Explanation: This is a second-order non-homogeneous linear differential equation, where: - \( y'' \) represents the second derivative of \( y \) with respect to \( x \). - \( 9y \) is a term where the function \( y \) is multiplied by 9. - The right-hand side, \( 2x^2 e^{3x} + 5 \), is the non-homogeneous part of the equation. - \( 2x^2 e^{3x} \) represents a term with a polynomial of degree 2 in \( x \) multiplied by an exponential function. - The constant 5 is an external forcing term. This equation might be analyzed or solved to find \( y(x) \), the function describing a particular physical or theoretical system influenced by the given non-homogeneity. Solutions typically involve finding both the complementary solution (related to the homogeneous equation) and a particular solution for the non-homogeneous part.
Expert Solution
Step 1: Given:

Calculus homework question answer, step 1, image 1

steps

Step by step

Solved in 4 steps with 7 images

Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
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: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning