Question

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Solve the initial value problem below using the method of Laplace transforms. y'' + 4y' - 12y = 0, y(0) = 2, y' (O) = 36 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. y(t) = 0 (Type an exact answer in terms of e.)

Answer 1

Solution Recall the properties of Laplace bansform hCf)(5) of the function f(t) i) unearity llaft g) (5) & L (f) GH L(g) (G) 11) L ( fr ) (9) SMLCL) (s) - 5 fo-5"?f?@- ni n-2 s f one CO2 Ys-a azo का -36 >(*) hi ) Lleat) (5) consider the initial value problem, y" + 4y - 12y=0 yo) = 2, y'CO) = 3 Apply Laplave transform bolto sides of CM) > Lly" + 4y -12%) = 2(0) a high) + 4 Lly) -12 Lly) - 0. (by linearity of L(f) Ś Lly) (5) - 54(0)- y/co) + 4 +4[s Lly)cs)-4601 12 LCY760 from ] z 5² LC4) (5) - Sx2 - 36 +4 [SLCy) (5) – 2] -12 ()() = 0 2 Let Lly 2052 Y(52 3² Yes2 - 25 - 36 + 4 Sys) - 8-12 y =0 YCS) [5²44 4.4+25 价 +45-12] Y(s) 44 tas 5+45-12 (Y)(0) 쿠 Y CE) 44+25 s+45-12

Here 4 4 + 25 s 44+25 +45-12 (3+6) (S-2) = by partial fraction s-2 StG 44+25 & I'l G s-2 S2445-12 StG I (9) -164) (bg loncorselen q I s. 1 () - 426tc) 6. p2t_ 4 (from ii)] -6t -) gct) 2) consider the increal value problem y"-sy' 4119 = 8 go) = 4, y)=1 3t =13 La Apply Laplace bansferon on both sides of G2 > h(y"-sy +174 VEE L(8e3t) (s) - Llyn2348 (g/2531 Lly)0)=8 ble3t) 62 (by si LC4765) -5 410) – ylco) - 8 [SLCY)(5) - Y67] +1 +18 Lly) cs) from lii)¢ Let LCY268)= Y(52 ) 3*y (5)-5.24 - 13 - 8[sx(s) -47 +14LY(S=_3 s-3 s-3 2 sy(s) – 43-13-8sy(s) +32 +1+4 (5) 5-3 2_7(52$?-85411] 17 +45-19 S s-3

719) 86-2 445-19 2 S-85 tlf Yct 2 I'CY (5) 2 / -3 +49-19 3² 854 If 8 +(45-19) (5-3) 15-3) CS-85411) 2 Hore 8+(45-19) CS-3) (5-3)(5285411) 8 + 45-125-19 5+5=1 (5-3) (5?-85+17) 48-315_465 (5-3) (52-85411) 4t S-3 S-8871 2 4 + s-3 S2-85+16+ = 4 . I t I S-3 (3-4)² +12 Recall that 1) Le at fck))(5) = 1(£)(5-a) Ll sin bt) s² +62 Anleat sunbl) (5-a) ² 46² b Thus, I (419) 4. It 5.-3 J 22 I'T (5-4) 11 4[ (1) + I set 2) (5-45+1 5-3 = 4at teet sint 4. e3t test sint u yct)