Question

where h is the Use the Laplace transform to solve the following initial value problem: y"+y + 2y = h(t – 5), y(0) = 2, y(0) = -1, Heaviside function. In the following parts, use h(t – c) for the shifted Heaviside function he(t) when necessary. a. First, take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation and then solve for L{y(t)}. L{y(t)}(s) = b. Express the solution y(t) as the sum of the solution the corresponding homogeneous initial value problem and a convolution integral. g(t) = + وت dw

Answer 1

consider given ini following initial value problem Y"+y+zy = h17-55) ,Y 20)=2 y'o) =-) Where h is heaviside function have to solve giver zup using laplace transform consider differential equation y "toy try hot-5) - iwe 11 and taking lead a taking laplace transform of solving for LPylts} :2py" tay't zy} = 24 hits Lfyly polfyltzeff} = oss ) S (² uph(z-as} = mas S ... ]۔ séhef yang .; E 28'45} + {2 SY0) – you] ++ [ 5228 yet} -yco] 72 24 y(t)} ess 12 S

7 C L{fich} 8 lffCHY -FCO) & Lf fictly = 3 lof fctly sfog -fico) {selfy y (ts} -3(2) + 1] + [Selfyty-2] e falegeti 55 = e S -55 e S -55 e ..2 sts 0-25 +7 25+ = s .: Lf44] o (s +3+3) —25+1 -2 -2 fycts} (3 7512) Lf y(+1} ($+s+2) – (25+1) Alfy en} (3²4s+2) --S e s 55 (25+1) saa t :.Lf yets} s(8457 (28+)) s27512 S 55 e 2 + (S) . {ff]hg 7 s(sista (254) PD) (8²45+2) Now coe taking inverse laplace pensform of 2fy CH}(s) to find yets 27f Lş y cts}} 278 -55 e Yt= 十 (251)).7 s²x512 S (SPS+2) - YCH = 25$ sess fost {re (25+)22 31512 S (545+2)

Firstly we find LY ē55 For this ses²4 S121 convolution theorem we use let F(S)= G(S) = 255 : HP FCs) y = 4*3*2} = En} spes*****? = 21 (57163 + 5 7 = ) مریم تا (8+1/2² - + (7) = that its s 1 (J2² - oht (H/F (59)} = Et Llf F(s)}) sin (JFK+) ( 2"} FCs)} = zaht crap 2012 (8) sin(JFt) aj7 If Alto orosz & 27$ 60s)} = Ufes}= h(2-5) = hg(t) .: by convolution theorem 21 f F(s)} & HfG(s)} = 9(7) then Ehf FC) 669Strcu f(4).964-4) dy = f*g Herre Llf FCs)}= I sin (It) =f(t) 44"} Ges)} = h(U) = nyt Rez & JF gG)

124 sin t_164 UfFG)GCS)} ve sin(Jxy). 55(7-4) dy .2437 spela e sin ( 57 ) hos (t-udlu E's frey ? S (1512) :. Now We find LH consider (2011) (Rest2) 102438443) (8711 712) S+ 1/2) 52412 (S+½)²+(II) 2 = 2x2 Lt بے - axelet (":48F65-9)} scat 444FCs)}) cos (ET) :17} etno se 224 cor ( 4sea}sted in sin(y) No (7-4) dy + zelkt co( t) potting He ses4sez we 94 YH) - 1/24 JE 2 1-y(t) = zeros (II t) it J7 (Ity) by study