TMA4130 MATEMATIKK 4NLecture 10: Transforms of Derivatives andIntegrals. ODEs

Elisabeth KöbisSeptember 22/24, 2021

Plan for the day

After today’s lecture, you should be familiar withI the Laplace transform of derivatives and integralsI solving ODEs with the Laplace transform.

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Solving linear ODEs and related initial valueproblems

The Laplace transform is a method of solving ODEs andinitial value problems. The crucial idea is that operations ofcalculus on functions are replaced by operations of algebraon transforms. Roughly, differentiation of f(t) willcorrespond to multiplication of L(f) by s and integration off(t) to division of L(f) by s. To solve ODEs, we must firstconsider the Laplace transform of derivatives.

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Theorem: Laplace Transform of DerivativesThe transforms of the first and second derivatives of f(t)satisfy

L(f ′) = sL(f)− f(0) (1)L(f ′′) = s2L(f)− sf(0)− f ′(0). (2)

Formula (1) holds if f(t) is continuous for all t = 0 andsatisfies the growth restriction1 and f ′(t) is piecewisecontinuous on every finite interval on the semi-axis t = 0.Similarly, (2) holds if f and f ′ are continuous for all t = 0 andsatisfy the growth restriction and f ′′ is piecewise continuouson every finite interval on the semi-axis t = 0.

1∃ M,k s.t. ∀t = 0 : |f(t)| 5 Mekt.

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Theorem: Laplace Transform of the Derivative f (n) ofAny OrderLet f, f ′, . . . , f (n−1) be continuous for all t = 0 and satisfy thegrowth restriction2. Furthermore, let f (n) be piecewisecontinuous on every finite interval on the semi-axis t = 0.Then the transform of f (n) satisfiesL(f (n)) = snL(f)− sn−1f(0)− sn−2f ′(0)− . . .− f (n−1)(0)).

2∃ M,k s.t. ∀t = 0 : |f (i)(t)| 5 Mekt.

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ExampleLet f(t) = t sinωt. Find L(f ′′) and L(f).

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ExampleLet f(t) = cosωt and g(t) = sinωt. Find L(f) and L(g).

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Theorem: Laplace Transform of IntegralLet F (s) denote the transform of a function f(t) which ispiecewise continuous for t = 0 and satisfies a growthrestriction3. Then, for s > 0 and t > 0,L{∫ t

0f(τ)dτ

}=

1

sF (s), thus

∫ t

0f(τ)dτ = L−1

{1

sF (s)

}.

3∃ M,k s.t. ∀t = 0 : |f(t)| 5 Mekt.

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ExampleFind the inverse of 1

s(s2 + ω2)and 1

s2(s2 + ω2).

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Differential Equations, Initial Value ProblemsLet us now discuss how the Laplace transform methodsolves ODEs and initial value problems. We consider aninitial value problem

y′′ + ay′ + by = r(t), y(0) = K0, y′(0) = K1,

where a and b are constant.r(t) . . .y(t) . . .

In Laplace’s method we perform three steps:Step 1. Setting up the subsidiary equation.Step 2. Solution of the subsidiary equation by algebra.Step 3. Inversion of Y to obtain y = L−1(Y ).

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Step 1. Setting up the subsidiary equation.

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Step 2. Solution of the subsidiary equation byalgebra.

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Step 3. Inversion of Y to obtain y = L−1(Y ).

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ExampleSolve

y′′ − y = t, y(0) = 1, y′(0) = 1.

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Comparison with the Usual MethodExampleSolve

y′′ + y′ + 9y = 0, y(0) = 0.16, y′(0) = 0.

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Advantages of the Laplace Method

1. Solving a nonhomogeneous ODE does not require firstsolving the homogeneous ODE.2. Initial values are automatically taken care of.3. Complicated inputs r(t) (right sides of linear ODEs) canbe handled very efficiently, as we show later.

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Shifted Data ProblemsExampleSolve

y′′ + y = 2t, y

(1

4π

)=

1

2π, y′

(1

4π

)= 2−

√2.

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Conclusion

F (s) := L(f) :=∫ ∞0

e−stf(t)dt

What we have learned todayI Laplace Transform of the derivative of a functionI Laplace Transform of the integralI Solving ODEs and Initial Value Problems using LaplaceTransformI Shifted data problems

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Next Lecture

Chapter 6.3, 6.4 in KreyszigI Unit Step Function (Heaviside Function)I Second Shifting Theorem (t-Shifting)I Dirac’s Delta Function

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ReferencesThe material of this lecture was based on Chapter 6.2 of thebookAdvanced Mathematical Engineering by Erwin Kreyszig (JohnWiley & Sons, 10th edition, 2011)and Chapter 6 inDifferential Equations Demystified by Steven G. Krantz(McGraw-Hill, 2005).Moreover, we recommend the lecture notes by MortenNome (in Norwegian), who taught the 2019 edition of thiscourse. You can download Lecture 1 of Morten’s lecturenotes collection here:https://www.math.ntnu.no/emner/TMA4125/2019v/notater/01-laplacetransform.pdf

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