7.2 inverse Laplace Transforms, and application to DEs a lesson for - - PowerPoint PPT Presentation

7.2 inverse Laplace Transforms, and application to DEs

a lesson for MATH F302 Differential Equations Ed Bueler, Dept. of Mathematics and Statistics, UAF

March 22, 2019 for textbook:

• D. Zill, A First Course in Differential Equations with Modeling Applications, 11th ed.

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recall the definition

• the Laplace transform of a function f (t) defined on (0, ∞) is

L {f (t)} = ∞ e−stf (t) dt

• this is well defined for s > c if f (t) has exponential order c:

|f (t)| ≤ Mect

• the result of applying the Laplace transform is a function of s:

L {f (t)} = L {f } (s) = F(s) ← − all mean the same

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the Laplace transform strategy

ODE IVP for y(t) algebraic equation for Y (s) Y (s) = . . . y(t) = . . . L solve for Y L−1

• ld method
• §7.2: practice with L−1 then practice the whole strategy

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bring a table to the party

• on page 282 of book
• this table is pathetic! better one soon . . .

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first L−1 example (like §7.2 #5)

• exercise 1. use algebra and a table of Laplace transforms:

L−1 (s − 1)3 s4

• =

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L−1 example like §7.2 #11

• exercise 2. use algebra and a table of Laplace transforms:

L−1

• 5

s2 + 36

• =

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L−1 example like §7.2 #18

• exercise 3. use algebra and a table of Laplace transforms:

L−1 s + 1 s2 − 7s

• =

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not actually a better table

• compare Theorems 7.1.1 and 7.2.1
• they say the same thing!

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actually a better table

• this substantial table will be printed on your quiz/exam

L {1} = 1 s L {t} = 1 s2 L

• tn

= n! sn+1 L

• t−1/2

= √π s1/2 L

• t1/2

= √π 2s3/2 L

• tα

= Γ(α + 1) sα+1 L

• eat

= 1 s − a L {sin(kt)} = k s2 + k2 L {cos(kt)} = s s2 + k2 L {sinh(kt)} = k s2 − k2 L {cosh(kt)} = s s2 − k2 L

• teat

= 1 (s − a)2 L

• tneat

= n! (s − a)n+1 L

• eat sin(kt)
• =

k (s − a)2 + k2 L

• eat cos(kt)
• =

s − a (s − a)2 + k2 L {t sin(kt)} = 2ks (s2 + k2)2 L {t cos(kt)} = s2 − k2 (s2 + k2)2 L

• eatf (t)
• = F(s − a)

L {U(t − a)} = e−as s L {f (t − a)U(t − a)} = e−asF(s) L

• f (n)(t)
• = snF(s) − sn−1f (0) − · · · − f (n−1)(0)

L

• tnf (t)
• = (−1)s dn

dsn F(s) L t f (τ)g(t − τ) dτ

• = F(s)G(s)

L {δ(t)} = 1 L {δ(t − t0)} = e−st0 9 / 18

L−1 example like §7.2 #23

• exercise 4. use algebra and a table of Laplace transforms:

L−1

• s

(s − 3)(s − 4)(s − 6)

• =

s (s−3)(s−4)(s−6) = 1 s−3 − 2 s−4 + 1 s−6 10 / 18

L−1 example like §7.2 #25

• exercise 5. use algebra and a table of Laplace transforms:

L−1

• 1

s3 + 7s

• =

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transform of first derivatives

• exercise 6. suppose F(s) = L {f (t)}. use the definition of the

Laplace transform to show: L {f ′(t)} = s F(s) − f (0)

• actually we showed this on §7.1 slides
• what assumptions did we make about f (t)?

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transform of second derivatives

• exercise 7. suppose F(s) = L {f (t)}. show:

L

• f ′′(t)
• = s2F(s) − s f (0) − f ′(0)
• in the table you’ll have in hand during quizzes/exams:

L

• f (n)(t)
• = snF(s) − sn−1f (0) − · · · − f (n−1)(0)

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like §7.2 #39

• exercise 8. use Laplace transform to solve the ODE IVP:

y′′ − 5y′ + 4y = 0, y(0) = 1, y′(0) = 0

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the old way

• exercise 9. solve without Laplace transform:

y′′ − 5y′ + 4y = 0, y(0) = 1, y′(0) = 0

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like §7.2 #41

• exercise 10. use Laplace transform to solve the ODE IVP:

y′′ + y = √ 2 cos( √ 2t), y(0) = 0, y′(0) = 3

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like §7.2 #41, cont.

y(t) = 3 sin(t) + √ 2 cos(t) − √ 2 cos( √ 2t)

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expectations

• just watching this video is not enough!
• see “found online” videos and stuff at

bueler.github.io/math302/week11.html

• read section 7.2 (and 7.1 and 7.3) in the textbook
• do the WebAssign exercises for section 7.2

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