Question

Use the Laplace transform technique to solve the following IVP (no credit will be given if you use another technique). (15 pts) y" - y' - 6y = 0; y(0) = 2, y'(0) = -1

Answer 1

Q solve 2 - 1 y lola by lupluce transform y" - y - by = 0 -0 with yrol know that Lig'll} s Llyes - ylo) =s Llyce) - 2 soll we 2 Lly"() 8² Llyce sylul ay'lo) = 3 = 8² L {y (t} Lleat) 11 4 sta from o - O y" (t/ - y'lt) - byle/ on applying L-transform Lloy"(t)} - Lly'llls - 6 Lly (1}=0 & 3 using @ → [ L {y141 - 28+] - [ s li'yle} - 2] 1 -6 Llylt I

-> (32-8-6) Ly1) = ls - 3 (8+2)(8-3) Lytt)) Lly(t) = 28-3 5 L {y (th 23 - 3 (5+2) (8-3) By bantial fraction B 28-3 + = 8-3 S+2 (3+2) (6-3) B(8+2) A (8-3) + 23-3 = = (A+B) + (-3A+ 2B/ on comparing A+B= 2 3 3A + 2B = on solving A B = 3/5 A = 7/s

therefore 3 28 - 3 7 518+2) -O Il + 518-3) (3+2)/3-3) the froom ② & 6 Lly(t)} = + 3 518-3) 518+2) L transform applying L y(t) } *6*2) + "63) lesing -21 3t e 3 t y(t) = Ie- m/h 5 sulm which is required