Question

Solve the initial value problem below using the method of Laplace transforms. y'' - 12y' +45y = 39 e 4t, y(0) = 3, y'(0) = 15 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. y(t) = (Type an exact answer in terms of e.)

Answer 1

Quee ylizy' +484 = 3ge at 4(0)- ay'(o)=15 dy't sees)-yeo) mering R$ "} o s2 Yee) -s Yeo) -y.com - 5-4 Take saplace transform of equo 3-4. 324 (3) - SYCO) -4'o) – 12 (SY (6) – 4 (0)) + 4576) - 39 s-4 5-4 35-21 s?y (s) - 35 -15 -125Y (8) + 12x3 +454S) = 3 (8-125 +45) 409. = 39 +35 -21 Yes) sc-, 25+45) + 125445 - 39+ (35 -21) (5-4) (5² - 125 +45) (5-4) - 39+ 35² – 125-213 + 84 (5-4) (5²-125+45) Y(s) = 35² _ 335+ 123 (5-4) (5²-125 +45) Use partial faith. 7 (6) = A + BstC 5- 4 52-125+45 352-335+123 = A[SL-125+457 +es+e) (5-4) 35% – 335 +123 = A (S2-125+45) + Bs? - 485 +(5-4C Comparing coefficients. A+B=3 -12A - 4B+C = -33 45A-4C = 123 2 sole

A Bro, Ca 3, no YOS) 2 3 S-4 + 2 125 +45 - Set 125 +36-36405 Yn = 3 + cajet 52 3 + 3 (6-6)2 32 S-4 ylt) = 3e4 on37 ) 3 e 4th +zost! B (721 y(t)= 3e 4t test sinat

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