To solve an initial value problem using the Laplace transform takes three steps. 1. Take the Laplace transform of both sides of your ODE. This transforms your ODE into an ALGEBRAIC equation! 2. Solve the resulting algebraic equation for Y(s) 3. Take the inverse Laplace transform. Your result is the solution, y(t). We have had some practice finding the Laplace transform and inverse Laplace transform of functions. However, we haven't had any practice taking the Laplace transform of derivatives (y', y'', y''',...), which will be part of step 1. The Laplace transform of the nth derivative of f(t) is given by L{f(n) (t)} = s"F(s) – s" - 'f(0) – s* -?f'(0) - sf(n- 2) (0) – f(n – 1) (0) ... Let's look specifically at the Laplace transform of y'(t),y'"(t), and y'"'(t). L{y} = Y L{y'} = sY – y(0) L{y''} = s?Y – sy(0) – y'(0) L{y''} = s³Y – s'y(0) – sy'(0) – y''(0) Now let's have you try to use these formulae. Find the inverse laplace transform of y'(t),y'(t), and y'''(t) given that y(0) = – 10, y'(0) = 5, y'"(0) = 13 L{y'} = L{y''} = L{y'''} : Each of your answers should contain the variables Y and s. This question is case sensitive.
To solve an initial value problem using the Laplace transform takes three steps. 1. Take the Laplace transform of both sides of your ODE. This transforms your ODE into an ALGEBRAIC equation! 2. Solve the resulting algebraic equation for Y(s) 3. Take the inverse Laplace transform. Your result is the solution, y(t). We have had some practice finding the Laplace transform and inverse Laplace transform of functions. However, we haven't had any practice taking the Laplace transform of derivatives (y', y'', y''',...), which will be part of step 1. The Laplace transform of the nth derivative of f(t) is given by L{f(n) (t)} = s"F(s) – s" - 'f(0) – s* -?f'(0) - sf(n- 2) (0) – f(n – 1) (0) ... Let's look specifically at the Laplace transform of y'(t),y'"(t), and y'"'(t). L{y} = Y L{y'} = sY – y(0) L{y''} = s?Y – sy(0) – y'(0) L{y''} = s³Y – s'y(0) – sy'(0) – y''(0) Now let's have you try to use these formulae. Find the inverse laplace transform of y'(t),y'(t), and y'''(t) given that y(0) = – 10, y'(0) = 5, y'"(0) = 13 L{y'} = L{y''} = L{y'''} : Each of your answers should contain the variables Y and s. This question is case sensitive.
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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Transcribed Image Text:To solve an initial value problem using the Laplace transform takes three steps.
1. Take the Laplace transform of both sides of your ODE. This transforms your ODE into an ALGEBRAIC
equation!
2. Solve the resulting algebraic equation for Y(s)
3. Take the inverse Laplace transform. Your result is the solution, y(t).
We have had some practice finding the Laplace transform and inverse Laplace transform of functions.
However, we haven't had any practice taking the Laplace transform of derivatives (y', y'', y''',...),
which will be part of step 1. The Laplace transform of the nth derivative of f(t) is given by
L{f(m) (t)} = s"F(s) – * -'f(0) – s" -?f'(0) –
sf(n- 2) (0) – f(n – 1) (0)
...
Let's look specifically at the Laplace transform of y'(t),y'"(t), and y'"'(t).
L{y} = Y
L{y'} = sY – y(0)
L{y''} = s?Y – sy(0) – y'(0)
L{y''} = s³Y – s'y(0) – sy'(0) – y'"(0)
Now let's have you try to use these formulae. Find the inverse laplace transform of y'(t),y''(t), and
y'''(t) given that y(0) = – 10, y'(0) = 5, y''(0) = 13
L{y'} =
L{y''} =
L{y'''} :
Each of your answers should contain the variables Y and s. This question is case sensitive.
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