Question

please solve with steps and explain thanks

Question 5 Given the differential equation y'' + 5y' + 4y = 0, y(0) = 2, y'(0) = 1 Apply the Laplace Transform and solve for Y(8) = L{y} Y(8) = Now solve the IVP by using the inverse Laplace Transform y(t) = L-'{Y(s)} g(t) =

Answer 1

Hence, | Ht) = 3 et eht y" + sy'+4y=0 25 +11 25 +11 he -=[est ;(+]+s[ēstſsct) dt -f-sever 149. fa- vua o tem probationem . +3.1 The Ho) = 2 = 2; 7 (0) = 1 Taking the zaplace Bransform from both sides, we set L[7.7+5L [] +4L[2] =0 " Laplace fromsform in lineas 0 -/-28 +5 2 [81t)] -10 +55 L [atts] and L[0]=0. + 4 L[4+)) =0 L[a"] = ſést y"64) at 5+55+4) - [346)] ēst ſo"(t)dt - S-sére j(t) at L [20] 8 + 55+4 dt) Hence, Y (s). = [0-70)] + s[ēst, yu) tsjes = -1 +5(0-4) + ŚL [*Lt)] .. L[414] - Y(5) - (s+4)(3+1) 584 * 5+1 --1-25 +5 L [960)] (5+4) (5+1) | L[2(t)} = Sest ý (t) at (A+B) +(+48) (5+49(5+1) est S7 (4) dt -S-sēstyle) at [zke. 764) + Sest yet) at =[0-901] +5 L [3lt)] :: L [3l+}] - $ty -2 +SL (4) Taking inverse Laplace transformation from both sides, weser -Z' [sku] +3 ="[ ] ght + 3 et 25 +11 А B B. As + A + BS +4B A+B=2 A+ 4B = 11 +s fest 3B 9 i B = 3 A = -1 5+/ Y+) = e CS Scanned with CamScanner