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)=∫∞0e−stf(t)dt.
- t
- te−t
- sinhbt
- e2t−3et
- t2
2. Use the table of Laplace transforms to find the Laplace transforms of the following functions.
- coshtsint
- sin2t
- cos22t
- cosh2t
- tsinh2t
- sintcost
- sin(t+π4)
- cos2t−cos3t
- sin2t+cos4t
3. Show that ∫∞0e−stet2dt=∞ 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.
- f(t)={−t,0≤t<2,t−4,2≤t<3,1,t≥3.
- f(t)={t2+2,0≤t<1,4,t=1,t,t>1.
- f(t)={sint,0≤t<π/2,2sint,π/2≤t<π,cost,t≥π.
- f(t)={t,0≤t<1,2,t=1,2−t,1≤t<2,3,t=2,6,t>2.
5. Find the Laplace transform:
- f(t)={e−t,0≤t<1,e−2t,t≥1.
- f(t)={1,0≤t<4,t,t≥4.
- f(t)={t,0≤t<1,1,t≥1.
- f(t)={tet,0≤t<1,tet,t≥1.
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 ∫∞0e−stf(t)dt with respect to s under the integral sign.
7. Use the known Laplace transforms
L(eλtsinωt)=ω(s−λ)2+ω2and L(eλtcosωt)=s−λ(s−λ)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:
- Show that if limt→∞e−s0tf(t) exists and is finite then f is of exponential order s0.
- Show that if f is of exponential order s0 then limt→∞e−stf(t)=0 for all s>s0.
- 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 τ≥0) such that |g(t)|≤w(t) for t≥τ 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 lims→∞F(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 limt→0+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,∞).
- Prove: If the integral g(t)=∫t0e−s0τf(τ)dτ satisfies the inequality |g(t)|≤M(t≥0), then f has a Laplace transform F(s) defined for s>s0. HINT: Use integration by parts to show that ∫T0e−stf(t)dt=e−(s−s0)Tg(T)+(s−s0)∫T0e−(s−s0)tg(t)dt
- 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.
- 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.
- sinωtt(ω>0)
- cosωt−1t(ω>0)
- eat−ebtt
- cosht−1t
- sinh2tt
16. The gamma function is defined by
Γ(α)=∫∞0xα−1e−xdx,
which can be shown to converge if α>0.
- Use integration by parts to show that Γ(α+1)=αΓ(α),α>0.
- Show that Γ(n+1)=n! if n=1, 2, 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 t≥0. (We say in this case that f is periodic with period T.)
- Conclude from Theorem 8.1.6 that the Laplace transform of f is defined for s>0.
- Show that F(s)=11−e−sT∫T0e−stf(t)dt,s>0. HINT: Write F(s)=∞∑n=0∫(n+1)TnTe−stf(t)dt Then show that ∫(n+1)TnTe−stf(t)dt=e−nsT∫T0e−stf(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:
- f(t)={t,0≤t<1,2−t,1≤t<2,f(t+2)=f(t),t≥0
- f(t)={1,0≤t<12,−1,12≤t<1,f(t+1)=f(t),t≥0
- f(t)=|sint|
- f(t)={sint,0≤t<π,0,π≤t<2π,f(t+2π)=f(t)