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8.1E: Introduction to the Laplace Transform (Exercises)

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Q8.1.1

1. Find the Laplace transforms of the following functions by evaluating the integral F(s)=0estf(t)dt.

  1. t
  2. tet
  3. sinhbt
  4. e2t3et
  5. t2

2. Use the table of Laplace transforms to find the Laplace transforms of the following functions.

  1. coshtsint
  2. sin2t
  3. cos22t
  4. cosh2t
  5. tsinh2t
  6. sintcost
  7. sin(t+π4)
  8. cos2tcos3t
  9. sin2t+cos4t

3. Show that 0estet2dt= for every real number s.

4. Graph the following piecewise continuous functions and evaluate f(t+), f(t), and f(t) at each point of discontinuity.

  1. f(t)={t,0t<2,t4,2t<3,1,t3.
  2. f(t)={t2+2,0t<1,4,t=1,t,t>1.
  3. f(t)={sint,0t<π/2,2sint,π/2t<π,cost,tπ.
  4. f(t)={t,0t<1,2,t=1,2t,1t<2,3,t=2,6,t>2.

5. Find the Laplace transform:

  1. f(t)={et,0t<1,e2t,t1.
  2. f(t)={1,0t<4,t,t4.
  3. f(t)={t,0t<1,1,t1.
  4. f(t)={tet,0t<1,tet,t1.

6. Prove that if f(t)F(s) then tkf(t)(1)kF(k)(s). HINT: Assume that it's permissible to differentiate the integral 0estf(t)dt with respect to s under the integral sign.

7. Use the known Laplace transforms

L(eλtsinωt)=ω(sλ)2+ω2and L(eλtcosωt)=sλ(sλ)2+ω2

and the result of Exercise 8.1.6 to find L(teλtcosωt) and L(teλtsinωt).

8. Use the known Laplace transform L(1)=1/s and the result of Exercise 8.1.6 to show that

L(tn)=n!sn+1,n= integer.

9. Exponential order:

  1. Show that if limtes0tf(t) exists and is finite then f is of exponential order s0.
  2. Show that if f is of exponential order s0 then limtestf(t)=0 for all s>s0.
  3. Show that if f is of exponential order s0 and g(t)=f(t+τ) where τ>0, then g is also of exponential order s0.

10. Recall the next theorem from calculus.

Theorem 8.1E.1

Let g be integrable on [0,T] for every T>0. Suppose there’s a function w defined on some interval [τ,) (with τ0) such that |g(t)|w(t) for tτ and τw(t)dt converges. Then 0g(t)dt converges.

Use Theorem 8.1E.1 to show that if f is piecewise continuous on [0,) and of exponential order s0, then f has a Laplace transform F(s) defined for s>s0.

11. Prove: If f is piecewise continuous and of exponential order then limsF(s) = 0.

12. Prove: If f is continuous on [0,) and of exponential order s0>0, then

L(t0f(τ)dτ)=1sL(f),s>s0. HINT: Use integration by parts to evaluate the transform on the left.

13. Suppose f is piecewise continuous and of exponential order, and that limt0+f(t)/t exists. Show that

L(f(t)t)=sF(r)dr. HINT: Use the results of Exercises 8.1.6 and 8.1.11.

14. Suppose f is piecewise continuous on [0,).

  1. Prove: If the integral g(t)=t0es0τf(τ)dτ satisfies the inequality |g(t)|M(t0), then f has a Laplace transform F(s) defined for s>s0. HINT: Use integration by parts to show that T0estf(t)dt=e(ss0)Tg(T)+(ss0)T0e(ss0)tg(t)dt
  2. Show that if L(f) exists for s=s0 then it exists for s>s0. Show that the function f(t)=tet2cos(et2) has a Laplace transform defined for s>0, even though f isn’t of exponential order.
  3. Show that the function f(t)=tet2cos(et2) has a Laplace transform defined for s>0, even though f isn’t of exponential order.

15. Use the table of Laplace transforms and the result of Exercise 8.1.13 to find the Laplace transforms of the following functions.

  1. sinωtt(ω>0)
  2. cosωt1t(ω>0)
  3. eatebtt
  4. cosht1t
  5. sinh2tt

16. The gamma function is defined by

Γ(α)=0xα1exdx,

which can be shown to converge if α>0.

  1. Use integration by parts to show that Γ(α+1)=αΓ(α),α>0.
  2. Show that Γ(n+1)=n! if n=1, 2, 3,….
  3. From (b) and the table of Laplace transforms, L(tα)=Γ(α+1)sα+1,s>0, if α is a nonnegative integer. Show that this formula is valid for any α>1. HINT: Change the variable of integration in the integral for Γ(α+1).

17. Suppose f is continuous on [0,T] and f(t+T)=f(t) for all t0. (We say in this case that f is periodic with period T.)

  1. Conclude from Theorem 8.1.6 that the Laplace transform of f is defined for s>0.
  2. Show that F(s)=11esTT0estf(t)dt,s>0. HINT: Write F(s)=n=0(n+1)TnTestf(t)dt Then show that (n+1)TnTestf(t)dt=ensTT0estf(t)dt and recall the formula for the sum of a geometric series.

18. Use the formula given in Exercise 8.1.17b to find the Laplace transforms of the given periodic functions:

  1. f(t)={t,0t<1,2t,1t<2,f(t+2)=f(t),t0
  2. f(t)={1,0t<12,1,12t<1,f(t+1)=f(t),t0
  3. f(t)=|sint|
  4. f(t)={sint,0t<π,0,πt<2π,f(t+2π)=f(t)

This page titled 8.1E: Introduction to the Laplace Transform (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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