Given the differential equation y'' - 3y' - 4y = 0, y(0) = 2, y'(0) = -1 Apply the Laplace Transform and solve for Y(s) = L{y} Y(s) = Now solve the IVP by using the inverse Laplace Transform y(t) = L¯¹{Y(s)} y(t) =
Given the differential equation y'' - 3y' - 4y = 0, y(0) = 2, y'(0) = -1 Apply the Laplace Transform and solve for Y(s) = L{y} Y(s) = Now solve the IVP by using the inverse Laplace Transform y(t) = L¯¹{Y(s)} y(t) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Solving a Second-Order Differential Equation using the Laplace Transform**
Given the differential equation:
\[ y'' - 3y' - 4y = 0 \quad \text{with initial conditions} \quad y(0) = -2, \quad y'(0) = -1 \]
### Applying the Laplace Transform
First, we apply the Laplace Transform to the differential equation and solve for \( Y(s) = \mathcal{L}\{y\} \):
\[ Y(s) = \]
### Solving the Initial Value Problem (IVP)
Now, solve the initial value problem (IVP) by using the inverse Laplace Transform:
\[ y(t) = \mathcal{L}^{-1}\{Y(s)\} = \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d3733a5-5e9e-433b-b6d1-2cfec636672d%2F648aefbf-ad54-4781-9ea6-02495ae7e018%2Fbs4qte_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving a Second-Order Differential Equation using the Laplace Transform**
Given the differential equation:
\[ y'' - 3y' - 4y = 0 \quad \text{with initial conditions} \quad y(0) = -2, \quad y'(0) = -1 \]
### Applying the Laplace Transform
First, we apply the Laplace Transform to the differential equation and solve for \( Y(s) = \mathcal{L}\{y\} \):
\[ Y(s) = \]
### Solving the Initial Value Problem (IVP)
Now, solve the initial value problem (IVP) by using the inverse Laplace Transform:
\[ y(t) = \mathcal{L}^{-1}\{Y(s)\} = \]
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