LAPLACE TRANSFORM REVIEW SOLUTIONS

Transcription

1 LAPLACE TRANSFORM REVIEW SOLUTIONS. Find the Laplace tranform for the following function. If an image i given, firt write out the function and then take the tranform. a e t inh4t From #8 on the table: 4 L {inh4t} 2 6 Uing #2 on the table: L {e at f t} Y a L {e t 4 inh4t} b.5 in3t π/2 f t.5in3t π/2.5in3tcoπ/2.5inπ/2co3t.5co3t L {f t} Note: we do not ue #3 from the table t-hift becaue f t i not multiplied by a Heaviide tep function. c Function given in the following figure: Whenever you re given an image that you need to find the tranform of, it i firt helpful to write it a a piecewie function, and then tranform thi piecewie function into a ingle-line" form with Heaviide tep function we can take the Laplace tranform of: t 0 < t < f t < t < 2 0 t > 2 Now we can multiply each part of the piecewie function with a boxcar function to window" that part of the piecewie function to only it relevant interval: f t ut t + ut ut 2 CME 02 Spring 5-6 Tim Anderon

2 Finally, we hould rearrange thi to be in a form uch that we can apply #3 from the table t-hift: f t ut t + ut ut 2 t tut + ut ut 2 t t + ut + ut ut 2 t t + ut + ut ut 2 t t ut ut + ut ut 2 t t ut ut 2 L {f t} 2 e 2 e 2 2. Solve the following initial value problem: a y + 9y 0e t, y0 y 0 0 Apply initial condition: y + 9y 0e t 2 Y y0 y 0 + 9Y 0 + Now we need to do a partial fraction decompoition: 2 Y + 9Y 0 + Y Y A B +C A B + +C + A 2 + 9A + B 2 + B +C +C 2 A + B + B +C + 9A +C A B C Finally, ubtitute back in the partial fraction and take the invere tranform: 0 Y CME 02 Spring 5-6 Tim Anderon

3 yt e t co3t + 3 in3t b y + y 0.02t 2, y0 25, y 0 0 y + y 0.02t 2 2 Y y0 y 0 + Y Y Y 2 + Y Now, notice that we need to do a partial fraction decompoition on the firt fraction but not on the econd. We can already take the Laplace tranform of thi, o we do not need to decompoe it not that we could, anyway. And now we can find the final olution: A 3 + B 2 + C + D + E 2 + A B 0 C 25 D 25 E Y yt 2 t Find the invere Laplace tranform: a Y 2e e Y 2e e CME 02 Spring 5-6 Tim Anderon

4 2e e e 2 4 2e Uing #8 and #3 on the tranform table: yt ut inh2t ut 3inh2t 3 b Y +e2π Firt plit up the fraction, ince thi will make it eaier to handle the invere tranform: Y + e2π e2π Now, notice that we have both an -hift and a t-hift. When applying the hift, you firt apply the hift in the domain you are in, then apply the hift for the domain you are going to. So in thi cae, we need to take cae of the -hift, then take care of the t-hift. Y e2π yt e t int + ut + 2πe t+2π cot + 2π 4. Solve the initial value problem: where f t 8int for 0 < t < π and 0 for t > π. y + 9y f t, y0 0, y 0 4 We can ue tep function to window" the ine function to only be over 0 < t < π. We can thu rewrite f t with tep function, then olve the ODE: f t ut π8int y + 9y ut π8int 8int 8ut πint 8int 8ut πint π + π 8int 8ut πint πcoπ + inπcot π 8int + 8ut πint π 2 Y 4 + 9Y e π Y e π CME 02 Spring 5-6 Tim Anderon

5 Uing partial fraction: A A 8 B B Reubtitute thi fraction and take the invere tranform: Y e π e π e π yt int 3 in3t + ut π int π 3 in3t π in3t int + in3t + ut π int π 3 in3t π 5. Solve the initial value problem: y + 4y + 5y δt, y0 0, y 0 3 We can olve thi directly uing the Laplace tranform: y + 4y + 5y δt 2 Y y0 y 0 + 4Y 4y0 + 5Y e 2 Y 3 + 4Y + 5Y e Y e + 3 Y e We need to complete the quare for the denominator, then we can take the invere tranform: Y e e e e yt ut e 2t int + 3e 2t int CME 02 Spring 5-6 Tim Anderon

6 6. a Find the Laplace tranform: f t 2 te 3t Uing #9 on the Laplace tranform table: L {t n g t} n G n L {t g t} d d G g t 2 e 3t f t t g t F d L {g t} d d 2 d b Find the invere Laplace tranform: F cot π. Hint: d dx cot x. +x 2 We have to ue #9, but in the other direction thi time: F cot π F π + /π 2 π π F π π Uing #6 on the tranform table: and then uing #9 from the table: L { F } inπt t f t inπt f t inπt t 7. Solve y + 4y f t, y0 y 0 0 with f t defined by the following figure: CME 02 Spring 5-6 Tim Anderon

7 You firt need to expre f t a a piecewie function, then expre it uing Heaviide tep function: 0 < t < f t < t < 2 0 t > 2 f t ut ut ut 2 2ut + ut 2 Now, we can plug thi into our ODE and olve via Laplace tranform. y + 4y f t 2 Y y0 y 0 + 4Y 2e 2 Y + 4Y 2e 2ut + ut 2 + e 2 + e 2 Y e + e 2 Y 2e + e A B +C A 2 + 4A + B 2 +C 4A, 0 C, 0 A + B A 4 B Y 2e + e e yt 4 4 co2t 2ut co2t ut 2 4 co2t e CME 02 Spring 5-6 Tim Anderon