Use the Laplace transform to solve the given initial-value problem. y + 4yet, y(0) - 2 Step 1 To use the Laplace transform o solve the given initial value problem, we first take the transform of each member of the differential equation +4y=e³t. The strategy is that the new equation can be solved for (y) algebraically. Once solved, transforming back to an equation for y gives the solution we need to the original differential equation. {{0}+ We now recall the following, where a and b are constants. + £{4y)=L(0²³²) By Theorem 7.2.2.: . Using linearity of L: z(ay) = ax{v} • By Theorem 7.1.1.: (000) == 0 Applying these gives the following result. sz(y)-y(0)+( -sz(y)-y(0) 1)x(x) = dt
Use the Laplace transform to solve the given initial-value problem. y + 4yet, y(0) - 2 Step 1 To use the Laplace transform o solve the given initial value problem, we first take the transform of each member of the differential equation +4y=e³t. The strategy is that the new equation can be solved for (y) algebraically. Once solved, transforming back to an equation for y gives the solution we need to the original differential equation. {{0}+ We now recall the following, where a and b are constants. + £{4y)=L(0²³²) By Theorem 7.2.2.: . Using linearity of L: z(ay) = ax{v} • By Theorem 7.1.1.: (000) == 0 Applying these gives the following result. sz(y)-y(0)+( -sz(y)-y(0) 1)x(x) = dt
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
![Tutorial Exercise
Use the Laplace transform to solve the given initial-value problem.
y' + 4y = e³t, y(0) = 2
Step 1
dy
To use the Laplace transform to solve the given initial value problem, we first take the transform of each member of the differential equation + 4y = e³t. The strategy is that the new equation can be solved for {y} algebraically. Once solved, transforming back to an equation
dt
for y gives the solution we need to the original differential equation.
+ L{4y} = L{e³t}
We now recall the following, where a and b are constants.
= s£{y} − y(0)
By Theorem 7.2.2.:
• Using linearity of L:
• By Theorem 7.1.1.:
L{ay} = a£{y}
{b} = 5=b
Applying these gives the following result.
] ) £{v} =
SL{y} - Y(0) +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F402c8046-937e-4eba-941f-75445d382ed3%2F9e03a491-c005-45a9-8290-fa00940071ff%2Fihmvvnq_processed.png&w=3840&q=75)
Transcribed Image Text:Tutorial Exercise
Use the Laplace transform to solve the given initial-value problem.
y' + 4y = e³t, y(0) = 2
Step 1
dy
To use the Laplace transform to solve the given initial value problem, we first take the transform of each member of the differential equation + 4y = e³t. The strategy is that the new equation can be solved for {y} algebraically. Once solved, transforming back to an equation
dt
for y gives the solution we need to the original differential equation.
+ L{4y} = L{e³t}
We now recall the following, where a and b are constants.
= s£{y} − y(0)
By Theorem 7.2.2.:
• Using linearity of L:
• By Theorem 7.1.1.:
L{ay} = a£{y}
{b} = 5=b
Applying these gives the following result.
] ) £{v} =
SL{y} - Y(0) +
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
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