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PRACTICE PROBLEMS CHAPTER 6 AND 7

I. Laplace Transform

1. Find the Laplace transform of the following functions.

(a) ft=sin2tcos2t (b) ft=cos23t (c) ft=te2tsin3t (e) ft=t2u3t (f) t2-4t4, ift≥2 (g)

5,ift≥3

(h) ft= {0,ift,

0,ift≥2

(i) ft={ cost,ift4,

0,ift≥4

(j) ft= et,ift≥1

2. Find the inverse Laplace Transform:

(a) Fs=1 s1s2-1 (b) Fs=2s3s24s13 (c) Fs=e-3s s-2 (d) Fs=1e-2s s26

3. The transform of the solution to a certain differential equation is given byXs=1-e-2s

s21. Determine the solution x(t) of the differential equation.

4. Suppose that the functionytsatisfies the DE y''-2y'-y=1, with initial values,

y0=-1,y'0=1.Find the Laplace transform of yt

5. Consider the following IVP: y''-3y'-10y=1,y0=-1,y'0=2

(a) Find the Laplace transform of the solution y(t). (b) Find the solution y(t) by inverting the transform.

6. Consider the following IVP:y''4y=4u5t,y0=0,y'0=1

(a) Find the Laplace transform of the solution y(t). (b) Find the solution y(t) by inverting the transform.

7. A mass m =1 is attached to a spring with constant k =5 and damping constant c = 2. At the instant

t= the mass is struck with a hammer, providing an impulse p = 10. Also, x0=0and x'(0)=0. a) Write the differential equation governing the motion of the mass. b) Find the Laplace transform of the solution x(t). c) Apply the inverse Laplace transform to find the solution.

II. Linear systems

1. Verify that x=et

1

02tet1

1is a solution of the system x'=2-1

3-2xet1

-1

2. Given the system

x'=tx-yetz,y'=2xt2y-z,z'=e-t3tyt3z , define x, P(t) and ftsuch that the system is represented as x'=Ptxft

3. Consider the second order initial value problem:

u''2u'2u=3sint,u0=2,u'0=-1 Change the IVP into a first-order initial value system and write the resulting system in matrix form.

4. Are the vectors x1=

1 -1

1,x2=0

1

1and x3=1

1

1 linearly independent?

5. Consider the system x'=

-2-6

01x

Two solutions of the system are

x1=et -2

1 and x2=e-2t1

0(a) Use the Wronskian to verify that the two solutions are linearly independent.

(b) Write the general solution of the system.

6. Consider two interconnecting tanks as shown in the figure. Tank 1 initially contains 80 L (liters) of

water and 100 g (grams) of salt, while Tank 2 initially contains 20 L of water and 50 g of salt. Water

containing 15g/L of salt is poured into tank 1 at a rate of 3 L/m while the mixture flowing into tank

2 contains a salt concentration of 35 g/L and is flowing at a rate of 3.5 L/min. The mixture flows

from tank 1 to tank 2 at a rate of 5 L/min. The mixture drains from tank 2 at a rate of 6 L/min, of

which some flows back into Tank 1 at a rate of 2 L/min, while the remainder leaves the tank. Let Q1 and Q2, respectively, be the amount of salt in each tank at time t. Write down differential equations and initial conditions that model the flow process.

7. Suppose the systemx'=Axhas the general solutionxt=

x1t x2t x3t=c1et -2 1

0c2e-2t

1 0

1c3e-t

0 1

1 Given the initial condition x0=

1 1 -1,find x1t,x2t and x3t.8. Solve the IVP x'=Ax with A=1-3

0-2and x0=1

39. Solve the IVP x'=x2y

y'=4x3ywith x0=3,y0=0.

10. Suppose that A is a real3×3matrix that has the following eigenvalues and eigenvectors

-2, 1 1

1,1i,1-i

2

1,1-i,1i

2

1

Find a fundamental set of real valued solutions to the systemx'=Ax.

11. Solve the initial value problem x1'=x1-2x2,x2'=2x1x2,x10=0,x20=4 using the

eigenvalue method. Express the solution in terms of real functions only (no complex functions).

ANSWERS TO PRACTICE PROBLEMS CHAPTER 6 AND 7

I. Laplace Transform

1. (a) Using the double angle trigonometric identity, the functionftcan be rewritten as

ft=1

2sin4t.Thus L{ft}=2

s216 (b) Using the half angle trigonometric identity, the functionftcan be rewritten as ft=1 2 1 ss s236 (c) Using the property L{tft}=-F's with

Fs=L{e2tsin3t}=3

s-229yields s-2292

(d) ft=[t-710]u7t.Thus L{ft}=e-7sL{t10}=e-7s

1 s210 s (e) 2 s36 s29 s (f) ft=1u2tt2-4t3=1u2t [t-22-1]

Thus L{ft}=1

se-2sL{t2-1}=1 se-2s 2 s3-1 s (g) ft=t-u3tt-5=t-u3t[t-3-2]. Thus

L{ft}=1

s2-e-3sL{t-2}=1 s2-e-3s 1 s2-2 s (h)

ft=utt--u2tt-=utt--u2tt-2 Thus L{ft}=e-sL{t}-e-2sL{t}=e-s

s2-e-2s 1 s2 s (i)

ft=cost-u4tcost=cost-u4tcost-4 Thus

L{ft}=s

2s2-e-4ss

2s2 (j) ft=tu1t[et-t]=tu1t[et-11-t-1-1]Thus

L{ft}=1

s2e-sL{et1-t-1}=1 s2e-s e s-1-1 s2-1 s2. (a) Using PFD,

Fs=-1

4 1 s1-1 2 1 s121 4 1 s-1.Thusft=-1

4e-t-1

2te-t1

4et(b) F(s) can be rewritten as

Fs=2s3

s229-1 3 3 s229.

Thus ft=e-2t2cos3t-1

3sin3t(c) The inverse Laplace is

{1 s-2}=e2t. ThusL-1 {e-3s s-2}=u3te2t-3 (d) Fs=1 6 6 s26e-2s 6 6 s26thus L-1{Fs}=1 6sin6t1 1-u2tsint 4.

Ys=-s3

s2-2s-11 ss2-2s-15. (a)

Ys=1

ss-5s2-1 s2. (b) yt=-1

101

35e5t-13

14e-2t

6. (a)

Ys=1

s24e-5s 1 s-s s24. (b) yt=1

7. (a)

x''2x'5x=10t- (b) Xs=10e-s s22s5=5e-s2

s124 (c) xt=5ute-t-sin2t-=5ute-tsin2t

II. Linear Systems

1. Differentiating the given x yields x'=et

1

1=3et2tet

2et2tet Substituting x into the right hand side of the DE yields:

2-1

3-2et2tet

2tetet

1 -1=2et4tet-2tet

3et6tet-4tetet

-et=3et2tet

2et2tet=x'

2. x= x y z Pt=t-1et 2t2-1

03tt3 ft=0

0 e-t3. u' v'=01 -2-2u v0

3sint u0

v0=2 -1

4. c11

-1

1c20

1

1c31

1

1=0

0

0 yields c1c3=0

-c1c2c3=0 c1c2c3=0 The only solution is c1=c2=c3=0,thus the vectors are linearly independent.

5. (a)

Wx1,x2=∣-2ete-2t

et0∣=-e-t≠0Thus the two solutions are linearly independent and form a fundamental set. (b)xt=c1et -2

1c2e-2t1

06.

dQ1 dt=452Q2

202.5t-5Q1

80,Q10=100

dQ2 dt=122.55Q1

80-6Q2

202.5t,Q20=507. x1t=6et-5e-2t

x2t=-3et4e-t x3t=-5e-2t4e-t

8. xt=-2et

1

03e-2t1

1=-2et3e-2t

3e-2t9.

xt=2e-te5t

yt=-2e-t2e5t10. The first eigenvalue/eigenvector pair gives the solution: x1t=e-2t

1 1

1 The second eigenvalue/eigenvector pair gives the two solutions:

x2t=et costsint 2cost cost,x3t=et -costsint 2sint sint.11. xt=-4etsin2t yt=4etcos2tquotesdbs_dbs5.pdfusesText_10