a) Prove 2 {y }=$ £{y} -y (0) Using the integral definition of Laplace transform: 2 {FH)} = So f (t) e-st dt v=y' V= est - Need Integration by parts L {y} = 5 y'e "dt any dva-ser yout 1. - 3. - Sye" dt. 1. 6 2 {y'} = -1²0) = 550=² ye₁²³ 124 y(c) де а L {y} = s£ {y}-gcol using дидор prove £ {y" } = 5²4 - syco) - y'(o) L y the fact A J c) Find £ {y''} with the same pattern.
a) Prove 2 {y }=$ £{y} -y (0) Using the integral definition of Laplace transform: 2 {FH)} = So f (t) e-st dt v=y' V= est - Need Integration by parts L {y} = 5 y'e "dt any dva-ser yout 1. - 3. - Sye" dt. 1. 6 2 {y'} = -1²0) = 550=² ye₁²³ 124 y(c) де а L {y} = s£ {y}-gcol using дидор prove £ {y" } = 5²4 - syco) - y'(o) L y the fact A J c) Find £ {y''} with the same pattern.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
I managed to get the first part, but I am not quite sure how to use the deriviative concept I am given.
![### Educational Content: Laplace Transform of Derivatives
#### Problem Overview
**a) Prove:**
\[
\mathcal{L} \{ y' \} = s \mathcal{L} \{ y \} - y(0)
\]
using the integral definition of the Laplace transform:
\[
\mathcal{L} \{ f(t) \} = \int_0^\infty f(t) e^{-st} \, dt
\]
**Note:** Integration by parts is needed.
**Steps:**
1. Begin with:
\[
\mathcal{L} \{ y' \} = \int_0^\infty y' e^{-st} \, dt
\]
2. Use integration by parts:
- Let \( u = y', \, dv = e^{-st} \, dt \)
- Then \( du = y'', \, v = -\frac{1}{s} e^{-st} \)
3. Apply the formula:
\[
\left[ y e^{-st} \right]_0^\infty - \int_0^\infty (-s y) e^{-st} \, dt
\]
4. Resolve the limits and the resulting integral:
\[
\mathcal{L} \{ y' \} = -y(0) + s \int_0^\infty y e^{-st} \, dt
\]
5. Final form:
\[
\mathcal{L} \{ y' \} = s \mathcal{L} \{ y \} - y(0)
\]
**b) Using part (a) and the fact \( y'' = (y')' \), prove:**
\[
\mathcal{L} \{ y'' \} = s^2 \mathcal{L} \{ y \} - s y(0) - y'(0)
\]
**c) Find**
\[
\mathcal{L} \{ y''' \}
\]
**using the same pattern.**
*Explanation:*
This set of problems involves finding the Laplace transforms of derivatives. The solutions involve integrating by parts and using the recursive property of derivatives and their transforms. Part (a) builds the foundational equation that is used in part](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc77f544a-ee3e-4902-8958-66e53c68e4a2%2F80673a5d-35bb-4e49-aa4a-f43b023c5739%2Fozjtv3_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Content: Laplace Transform of Derivatives
#### Problem Overview
**a) Prove:**
\[
\mathcal{L} \{ y' \} = s \mathcal{L} \{ y \} - y(0)
\]
using the integral definition of the Laplace transform:
\[
\mathcal{L} \{ f(t) \} = \int_0^\infty f(t) e^{-st} \, dt
\]
**Note:** Integration by parts is needed.
**Steps:**
1. Begin with:
\[
\mathcal{L} \{ y' \} = \int_0^\infty y' e^{-st} \, dt
\]
2. Use integration by parts:
- Let \( u = y', \, dv = e^{-st} \, dt \)
- Then \( du = y'', \, v = -\frac{1}{s} e^{-st} \)
3. Apply the formula:
\[
\left[ y e^{-st} \right]_0^\infty - \int_0^\infty (-s y) e^{-st} \, dt
\]
4. Resolve the limits and the resulting integral:
\[
\mathcal{L} \{ y' \} = -y(0) + s \int_0^\infty y e^{-st} \, dt
\]
5. Final form:
\[
\mathcal{L} \{ y' \} = s \mathcal{L} \{ y \} - y(0)
\]
**b) Using part (a) and the fact \( y'' = (y')' \), prove:**
\[
\mathcal{L} \{ y'' \} = s^2 \mathcal{L} \{ y \} - s y(0) - y'(0)
\]
**c) Find**
\[
\mathcal{L} \{ y''' \}
\]
**using the same pattern.**
*Explanation:*
This set of problems involves finding the Laplace transforms of derivatives. The solutions involve integrating by parts and using the recursive property of derivatives and their transforms. Part (a) builds the foundational equation that is used in part
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