the ential equation 2y" + ty'- 2y 14, y(0) = y'(0) = 0. n some instances, the Laplace transform can be used to solve linear differential equations with variable mono THEOREM 7.4.1 Derivatives of Transforms If F(s) = L{f(t)} and n = 1, 2, 3, ..., then Left)} = (-1)"_5), ds" to reduce the given differential equation to a linear first-order DE in the transformed function Y(s) = L{y(t)}. Solve the first-order DE for Y(s). Y(s) =
the ential equation 2y" + ty'- 2y 14, y(0) = y'(0) = 0. n some instances, the Laplace transform can be used to solve linear differential equations with variable mono THEOREM 7.4.1 Derivatives of Transforms If F(s) = L{f(t)} and n = 1, 2, 3, ..., then Left)} = (-1)"_5), ds" to reduce the given differential equation to a linear first-order DE in the transformed function Y(s) = L{y(t)}. Solve the first-order DE for Y(s). Y(s) =
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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![**Consider the Differential Equation:**
\[ 2y'' + ty' - 2y = 14, \quad y(0) = y'(0) = 0. \]
In some instances, the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. Use Theorem 7.4.1.
**THEOREM 7.4.1 Derivatives of Transforms**
If \( F(s) = \mathcal{L}\{t^n f(t)\} \) and \( n = 1, 2, 3, \ldots \), then
\[ \mathcal{L}\{t^n f(t)\} = (-1)^n \frac{d^n}{ds^n} F(s), \]
to reduce the given differential equation to a linear first-order differential equation in the transformed function \( Y(s) = \mathcal{L}\{y(t)\}. \)
**Solve the first-order DE for \( Y(s) \):**
\[ Y(s) = \boxed{\phantom{answer\ here}} \]
**Then find \( y(t) = \mathcal{L}^{-1}\{Y(s)\} \):**
\[ y(t) = \boxed{\phantom{answer\ here}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9bf7f6f8-9470-47f0-a3a4-f3c052528216%2F830d8feb-bf56-41d1-aaa8-b6d89c17ef0a%2Flokizgs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consider the Differential Equation:**
\[ 2y'' + ty' - 2y = 14, \quad y(0) = y'(0) = 0. \]
In some instances, the Laplace transform can be used to solve linear differential equations with variable monomial coefficients. Use Theorem 7.4.1.
**THEOREM 7.4.1 Derivatives of Transforms**
If \( F(s) = \mathcal{L}\{t^n f(t)\} \) and \( n = 1, 2, 3, \ldots \), then
\[ \mathcal{L}\{t^n f(t)\} = (-1)^n \frac{d^n}{ds^n} F(s), \]
to reduce the given differential equation to a linear first-order differential equation in the transformed function \( Y(s) = \mathcal{L}\{y(t)\}. \)
**Solve the first-order DE for \( Y(s) \):**
\[ Y(s) = \boxed{\phantom{answer\ here}} \]
**Then find \( y(t) = \mathcal{L}^{-1}\{Y(s)\} \):**
\[ y(t) = \boxed{\phantom{answer\ here}} \]
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