Solve the initial value problem below using the method of Laplace transforms. y" - 2y' - 15y = 0, y(0) = - 1, y'(0) = 11 Click here to view the table of Laplace transforms. 1 Click here to view the table of properties of Laplace transforms.2 y(1) = (Type an exact answer in terms of e.) 1: Table of Laplace Transforms f(t) F(s) = 2{f)(s) 1 e at , S> 0 n! th , n = 1,2,... sn+ 1: 5>0 b sin bt S cos bt 52 + 2 520 nl eatth, n = 1,2,.. S - ajn+ 1 : $>a b e at sin bt (s- a)2 + 2 : > > a s - a e at cos bt (s- a)2 + 2 : $ >a 2: Properties of Laplace Transforms { {f + g} = {{f) + fig) {{cf) = cfff) for any constant c featf(t) (s) = 1(0)(s - a) f (f'} (s) = SI(f)(s) - f(0) ( (f'') ( 5 ) = 52 eff} (s ) - sf (0 ) - f' ( 0 ) e (f()} (s) = s" eff)(s) - s"-1f(0) - sn -2f'(0) -... - f(n-1)(0) { {in f(t ) } ( s ) = ( - 1jnan don {- 1( F 1 + F 2 ) = 1 - "( F 1 } + 2 - ( F 2 ) { { CF ) = CA-' (F )2. Solve the initial value problem below using the method of Laplace transforms. y" - 12y' + 52y= 20 e 4t, y(0) = 1, y'(0) = 8 Click here to view the table of Laplace transforms.3 Click here to view the table of properties of Laplace transforms. 4 y(t) = (Type an exact answer in terms of e.) 3: Table of Laplace Transforms f(t) F(s) = L(f)(s) 1 1 e at n! th , n = 1,2,... sn + 1, S>0 b sin bt 52 + 2 : 5>0 S cos bt 54+ 2: 5>0 n! eatth n = 1,2,... S - ajn + 1 : > >a b e at sin bt 2:5>a (s - a)- +b s- a edcos bt (s - a)2 + 62' 2 : 5>a 4: Properties of Laplace Transforms { {f + g) = {{f) + fig) {{cf) = cf (f) for any constant c {{e atf(t)} (s) = {{f)(s - a) { {f') (s) = SI(f)(s) - f(0 ) { {f'') (s) = $2 fif)(s) - sf(0) - f'(0) e (film) } (s ) = self (s ) - sn - 1f(0 ) - sn - 2f' (0 ) -... - ((n- 1)(0) { {th f (1 ) ( s ) = ( - 1) n ({ {f)(s)) dan 2- 1( F 1 + F 2 ) = 1 1 { F 1 } + 2-( F 2 ) 1 { CF ) = CA (F )3. Solve the initial value problem below using the method of Laplace transforms. y" + 4y = 20t- - 4t + 26, y(0) = 0, y'(0) = 1 Click here to view the table of Laplace transforms. 5 Click here to view the table of properties of Laplace transforms. y(t) = 1 5: Table of Laplace Transforms f ( t ) F(s) = {{f)(s) 1 1 1 e at S -a : S >0 n! th , n = 1,2,... sh+1 , $>0 h sin bt S cos bt eatth n = 1,2,... S - a)n + 1 : $>a edsin bt (s - a)2 + b2 : s 6- 2 e at cos bt (S-a) + 2 : 5>a 6: Properties of Laplace Transforms fff + g) = {{f) + {{g) { {cf) = cf {f) for any constant c featf(t)) (s) = {{f)(s -a) f (f') (s) = siff)(s) - f(0) ( (f'') ( 5 ) = $2 eff}(s) - sf( 0 ) - f' ( 0 ) e (f() } (s) = s" eff)(s) - sn-1f(0) - sn -2f'(0) - ... - f(n-1)(0) I { th f (1 ) ( s ) = ( - 1 ) 0 -( 2 (0 ( s ) ) 1 - 1 ( F 1 + F 2 ) = 2 -' ( F 1 ) + 2- ( F 2 } { { CF ) = CA (F )