please solve the problems I will attach with details

T 8. y(I): r: "\ ro. y(il: e:, * tr. )(/): cos3r

9. y(t) : 73 ll. y(tl: e-3'

/@ ,,,:,: sin5r

202 CHAPTER 5 'The Laplace Transform

Exercises 8-13 are designed to test the validity of Proposi- tion 2. L In each exercise. (i) cornpute 4(,y')(s) for the given function, and (ii) compute s 4(.v)(s) - )(0) for the given function. Com-

pare this result to that found in part (i) to verify that t\ 4(,r,')(.r): s4(.y)(s) *)(0).

39. y" I y' + 2y : e-' cos2t, y(0) : l, y'(0) : -1 40. y" l2y' + 5y : t2e-', y(0) : l, y'(0) : -2 41. y" *5y:3e-tcos4t,y(0) = -1,y'(0):) 42. Here is an interesting way to compute the Laplace trans-

form of cos ror. (a) Using only Definition 1.1, show that f,{sint}(s) :

l/(s2 + l). (b) Suppose that /(/) has Laplace transform F(s). Show

that

Llf (ar\lls) = j. (;) Use this property and the result found in part (a) to show that

^C{sina,lt}(s) : -3-.s"+@- (c) If /(r) : sin cr.r/, then /'(r) : @ cos @t . Use Proposi-

tion 2.1 to show that

f{rocos<ot}(s): ": "s'+ a' thus ensuring

J' 4,{cos ror}ts) : ---,-;.s-+o'

-FIn a manner similar to that proposed in Exercises 8-13, verify

-p the result of Proposition 2.4 for the functions defined in Exer- - cises l4-17.E.-F r+. 1.(r) :11

N' ro. y(r) : sin2r 15. .v(r) : s-2l

/fO )'(r) = rr + 3I + s In Exercises 18-25, use Propositions 2.1,2.4, and2.7 to trans- form the given initial value problem into an algebraic equation involving,C(y). Solve the resulting equation for the Laplace transform of v.,,A

" \).v'+ 3.v: 12. v(0) : -l19.y'-5y:"",.y(0):I F ro. )' +5y : t2 * 2r + 3.y(0) : 0 \ Zt. y' - 4y : cos2r. y{0} : -2

22. y" * 1l : sin4/, y(0) :0, y'(0) : l /@ y" *2y' *2y : co, 2t, y(0): l, y'(0) :0

?* y" I y' +2y : cos2t * sin3r, )(0) : -1, y'(0) : I 49 y" + 3y'r5y : t + e-t,y(o) : -1, y'(o) : o

In Exercises 26-29, we Proposition 2.12 to find the Laplace transform of the given function.

2-6. y(t) : e-t sin3t 27. y(t) : e2' cosZt 1Q,,r, : e-2t (2t + 3) 29. y(t) = e-'(t2 + 3t + 4)

In Exercises 30-33, use Proposition 2.14 to find the Laplace transform of the given function. 30. y(r): /sin3/ r,Qr(r): re-' 32. y(t) : t2 cos2t 33. y(r) : 12r2t '

- In Exercise s 3441, use the propositions in Section 2 to trans- -'rt1 fo.rn the given initial value problem into an algebraic equation (\ N involving LO). Solve the resulting equation for the Laplace j L transform of y. iIr. y'+2y: rsinI. y(o): t t/ S- 35. y' - y : t2e-z' ,y(0) : 0 \- fO. y' + ]l, : e-' sin 3r, l(O) : O Lr \ l' Sz. y' - 2y - e2' cos t, y(0) - -Z

=:\

@tn"ro. mafunctionis defined by

l-1a.1 : [- ,.'r' ' ,lt. 1a > 0). Jo

(a) Prove that f(1) : 1 (b) Prove that l(cvt 1) : af (cv). [n fact, if r is a positive

integer, show that f (r * 1) : n!. (c) Show that

Llt'l(s' : f'11 "'{u1 r '

Indeed, ifr is a positive integer, use this result to shou that,C{r"}(s) : nl/ s"+t.

44. Suppose that I is a continuous function for I > 0 and i. of exponential order. (a) If /(/) + F(s) is a transform pair, prove that

c[ [' rr,rar]t,r - F'''.lJu' | .r Hint: Let g(r) : Ll .f trlrlr. Then note that 8'(t) = f (t) and use Proposition 2.1 to compute 4{g'(t)}(s t

(b) Use the technique suggested in part (a) to find

38. y" * 4y : t2 sin4t, y(0) :0, y'(0) : -1 ",1 I I' lr,,rt+l)l'

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