REVIEW FOR MT3 ANSWER KEY MATH 373 SPRING 15 PROF. YOICHIRO MORI This list of problems is not guaranteed to be an absolutel complete review. For completeness ou must also make sure that ou know how to do all of the homework assigned in the course up to and including that due on April 8 and all the worksheet problems through week 13 with emphasis on weeks 1 13 but with the following qualifications and exceptions: Material from Sec. 9.1 is NOT TESTED on MT3. BUT material from Sec. 9.1 COULD BE TESTED on the final exam. Answers for the review problems will be posted b the middle of Week 14 on the course web page. Wednesda and Thursda of Week 14 will be entirel devoted to review for MT3. The table of Laplace transforms supplied with this review sheet is the same as will be supplied on MT3 and on the final. Note that Appendix PF of the textbook provides help with partial fractions. To stud under tpical test conditions here are the ground rules. You ma use matrix addition and multiplication functions on our calculator along with det and rref functions but ou must indicate clearl where and how ou used these calculator functions otherwise our answers would be considered unjustified and get little credit. Fancier calculator functions e.g. those finding eigenvalues and eigenvectors are not allowed to justif our work. All algebra and calculus work (except for the matrix functions specificall allowed above) must be performed b hand and shown clearl. On this answer ke we provide answers and hints of solutions but in general do not provide complete solutions. Problem 1 1 for t < 1 Let f(t) 1 for 1 t < Find L{f(t)} directl from the definition. for t. F (s) 1 e st dt 1 e st dt e st s t1 Problem t + e st s t t1 1 e s + e s s Find L{(t + 1) e 3t } using the table of Laplace transforms supplied with this review sheet. Date: Answer ke posted Apr. 9 15. 1
PROF. YOICHIRO MORI F (s) L{t e 3t }+L{te 3t }+L{e 3t } Solve the following IVP [ [ x 3 5 b an method learned in class. [ x Problem 3 (s + 3) 3 + (s + 3) + 1 s + 3 s + 8s + 17 (s + 3) 3 + e t [ 1 4 [ x() () [ 4 36 [ x [ 5e t + 3e 3t 3e 5t 4e t + 3e 5t Problem 4 Find the inverse Laplace transforms of 47s 17 s +8s+41 and s+7 (s+1) (s+). e 4t (47 cos(5t) 41 sin(5t)) 3e t + 5te t 3e t Problem 5 Find the inverse Laplace transforms of 37s+3 (s+5)(s +4s+13) and 37s+3 (s+3)(s+5)(s+6). 9e 5t + 9e t cos(3t) + e t sin(3t). 81e 5t 44 3 e 3t 199 3 e 6t Problem 6 We consider two brine tanks. Initiall: Tank A contains 15 gallons of water and 37 pounds of salt. Tank B contains 5 gallons of water and 43 pounds of salt. Starting at time t : Brine at a concentration 3 lb gal gal of salt is pumped at 5 min into tank A. Brine is pumped from tank A to tank B at a rate of 5 gal Brine at a concentration lb gal of salt is pumped at 7 gal min min. into tank B. Brine is dumped down the drain from tank B at 1 gal min. Note that the amount of water in each tank remains constant. The brine is kept well-mixed at all times. Let x and denote the amounts of salt in tanks A and B respectivel. Write down the IVP determining x and for all times t. You do not have to solve the IVP.
REVIEW FOR MT3 ANSWER KEY MATH 373 SPRING 15 3 [ x [ 5 15 5 15 1 5 [ x [ 15 + 14 [ x() () [ 37 43 (I did not reduce fractions to lowest terms so that ou could see m reasoning.) Decouple the sstem [ [ x 1 3 4 [ x Problem 7 [ [ + e t 1 x() () [ 1 3 4 [ 7 4 Firstl diagonalize the coefficient matrix A i.e. find invertible P and [ [ x u diagonal D such that AP P D. Secondl make the substitution P w and simplif until ou get a pair of decoupled IVP s for u and w. Then STOP. You do not have to complete the solution of the given IVP... [ 1 3 4 [ 1 [ [ u 1 w 3 4 [ [ u 1 w [ u w [ 1 [ 1 [ [ 1 [ u w u w [ 1 +e t [ 1 + e t [ 1 1 [ 1 [ u() w() [ u() w() [ 9 11 [ 7 4 Solve the IVP [ [ x 1 3 4 b using Laplace transforms. [ x Problem 8 [ 1 + [ x() () [ 7 4 sx + 7 X Y + 1/s sy 4 3X 4Y + /s (s 1)X + Y 7 + 1/s 3X + (s + 4)Y 4 + /s 7 + 1/s 4 + /s s + 4 7s 35 X s 1 (s + 1)(s + ) 3 s + 4 s 1 7 + 1/s 3 4 + /s Y s 1 4s 3s + 1 s(s + 1)(s + ). 3 s + 4 [ [ x 8e t + 1e t 8e t + 1 + 63. e t
4 PROF. YOICHIRO MORI Problem 9 Solve the following IVP using Laplace transforms: +.4 3 () 5 () 1. s Y 5s 1 +.4(sY 5) 3/s Y 5s + 3s 3 s (s +.4) 8t + 95 45e.4t Problem 1 Solve the following IVP using Laplace transforms: + 7 + 1 6H(t 5) () () 3. Answer s Y s + 3 + 7(sY ) + 1Y 6e 5s /s (s + 7s + 1)Y s + 11 + 6e 5s /s s + 11 Y (s + 3)(s + 4) + 6e 5s s(s + 3)(s + 4). ( 1 5e 3t 3e 4t + H(t 5) e 3(t 5) + 3 ) e 4(t 5) Make the substitution into the IVP [ [ x 35 6 + 6 3 [ x [ x Problem 11 [ [ 3 3 [ x() () [ u w [ 3 9 [ x () () [ 4 5 and simplif in order to get decoupled initial value problems for u and w separatel. When ou are done simplifing ou can stop. You do not have to solve the resulting IVP s for u and w. (This is not exactl like an problem in class so far but the decoupling method we have learned adapts with ver little change to this situation.) [ 1 [ [ [ 3 35 6 3 6 3 6 3 3 39 [ 1 [ [ [ 1 [ 3 3 33 3 4 /13 3 9 9 3 5 u + 6u u() 33/13 u () 7/13 v + 39v v() 9/13 v () /13. [ 7 /13.
REVIEW FOR MT3 ANSWER KEY MATH 373 SPRING 15 5 Problem 1 Let f(t) (t 3)H(t 1) + (t 1)H(t ) (t + t)h(t 4). Express f(t) as a piecewise-defined function and graph it. for t < 1 t 3 for 1 t f(t) t + t 4 for t 4 4 for t 4. (Look at the end of the answer ke for the graph.) Problem 13 Find the Laplace transform of t t for t < f(t) 7 + t for t < 4 8 for t 4. f(t) (t t )(H(t) H(t )) + (7 + t )(H(t ) H(t 4)) + 8H(t 4) ( t + t)h(t) + (t t + 7)H(t ) + ( t + 1)H(t 4). L{f(t)} L{ t + t} + e s L{(t + ) (t + ) + 7)} + e 4s L{ (t + 4) + 1} L{ t + t} + e s L{t + 7t + 13} + e 4s L{ t 8t 15} s 3 + 1 ( 4 s + e s s 3 + 7 s + 13 ) ( + e 4s s s 3 8 s 15 ). s Problem 14 Find the inverse Laplace transform f(t) of F (s) e s /s + e s /s + 5e 3s /s 4e 5s /s rewrite f(t) in piecewise form and sketch its graph. f(t) H(t 1) + (t )H(t ) + 5H(t 3) 4(t 5)H(t 5) for t < 1 1 for 1 t < t 3 for t < 3 t + for 3 t < 5 t + for t 5. (See the end of the review sheet for the graph.) Problem 15 Find the inverse Laplace transforms of the following: e 5s 5s 9 s + 14s + 53 e 7s (s 5)
6 PROF. YOICHIRO MORI Solution. e 7(t 5) (5 cos((t 5)) sin((t 5)))H(t 5) (t 7)e 5(t 7) H(t 7). Problem 16 Solve the following IVP using Laplace transforms: + 8 + 41 3H(t 3) () 3 () 5. s Y + 3s 5 + 8(sY + 3) + 41Y 3e 3s /s (s + 8s + 41)Y + 3s + 19 3e 3s /s Y 3s 19 s + 8s + 41 + 3e 3s s(s + 8s + 41) 1 5 e 4t (15 cos(5t) + 7 sin(5t)) ( 3 +H(t 3) 41 3 ) 5 e 4(t 3) (5 cos(5(t 3)) + 4 sin(5(t 3))). (kinda ugl numbers sorr about that) Problem 17 Solve the following sstem b using Laplace transforms or b decoupling. (Preferabl give both a tr.) [ [ x 3 1 + [ x [ [ x() () [ 1 [ x () () We just do decoupling (which is fairl eas). [ [ [ 3 1 1 1 4 A P D 1 1 [ [ [ [ P 1 1 1/ P 1 1 1/ 1 [ [ [ x 1 1 u 1 1 w u + 4u u() 1/ u () 1 w + w w() 1/ w () 1. u 1 cos(t) 1 sin(t) w 1 cos( t) + 1 sin( t) [ x u + w 1 cos(t) 1 sin(t) + 1 cos( t) + 1 sin( t) u w 1 cos(t) 1 sin(t) 1 cos( t) 1 sin( t).
REVIEW FOR MT3 ANSWER KEY MATH 373 SPRING 15 7 Problem 18 Compute the convolution e t t directl from the definition. e t t t e w w dw w e w e w + dw 1 4 e w w e w ( e w w dw e t 1 4 e w w ) wt e w e (t w) w dw e t t Problem 19 Use Laplace transforms to compute the convolution e t t. w et 4 1 4 t L{e t t} L{e t }L{t} 1 s 1 s 1 s (s ). 1 s (s ) A s + B s + C s 1 As +Bs(s )+C(s ) (A+B)s +( B+C)s C C 1/ B 1/4 A 1/4 Represent the solution of the IVP e t t et 4 1 4 t Problem + 14 + 53 f(t) () () as the convolution of f(t) with the impulse response function for this problem. The impulse response function is the solution of the given problem for f(t) δ(t) and thus equal to ( ) L 1 1 s 1 + 14s + 53 e 7t sin(t). Thus ( ) 1 e 7t sin(t) f(t) is the answer to this problem.
8 PROF. YOICHIRO MORI Table of Laplace transforms f(t) L{f(t)} F (s) af(t) + bg(t) e st f(t) dt al{f(t)} + bl{g(t)} af (s) + bg(s) 1 1/s t 1/s t n n!/s n+1 e at 1/(s a) te at 1/(s a) t n e at n!/(s a) n+1 sin bt cos bt e at sin bt e at cos bt b s + b s s + b b (s a) + b s a (s a) + b tf(t) F (s) e at f(t) F (s a) f (t) sf (s) f() f (t) s F (s) sf() f () H(t a) e as /s f(t a)h(t a) e as F (s) f(t)h(t a) e as L{f(t + a)} δ(t) 1 δ(t a) e as t f(w) dw F (s)/s f(t) g(t) t f(t w)g(w) dw F (s)g(s) Note: We write H(t) step(t) to abbreviate. (H is for Heaviside.)
REVIEW FOR MT3 ANSWER KEY MATH 373 SPRING 15 9 Figure 1. Graph for problem 1
1 PROF. YOICHIRO MORI Figure. Graph for problem 14