Question

1. Find the Laplace transforms of these functions: r(t) = tu(t), that is, the ramp function; Ae-atu(t); Be atu(t). 2. Determine the Laplace transform of f(t) = 50cos ot u(t). 3. Obtain the Laplace transform of f(t) = (cos (2t) + e 41) u(t). 4. Find the Laplace transform of u(t-2). 5. Find vo(t) in the circuit shown below, assuming zero initial conditions. IH F + 10u(i) 42 v. (1)

Answer 1

a D Giren Altatult) Steps: cet x = Ult) transform on both sides Applying Caplace x(3) = / Be t'alt) steps. Applying RIS) Caplace transform to ( toni sht! on both sides. (0-1) en 6 xit) Meat ult) stept let glt) - utt) Applying taplace transform on both sides. Y(s) = 1/2 steps xt Meat ylt) Applying Caplace transform on both sides XLS). A Yista) X13). A sta c) let alt)= Be at alt) step! let ylt) = alt) Applying taplace transform com Y(3) = / on both sides

steps on both sides Xit) = Be ylt) Applying Caplace transform X68) B Ys+a) xo) B sta - 2 let flt) = socoswt ult) on both sides Applying caplace Fls) = transform sos Plt) = Cosat Fls) s star Stor 3) f(t) = (cosat teat) utt) step): f(t) = Applying cosat ult) te HE ult) Caplace transform on both sides fls) = ser & st llosat) F(s) - $15+4) +8+4 S+H) (STH) Als) = a (sta+2) (8 + H) (S+4)

4 steps : 21+) - ult->) Det ylt)= alt) Applying Caplace transtorm on both sides Ys) - Ys wit) ult-2) Applying Caplace transform on both sides let-a)) = eas) step 2: -23 X(s) = XC) e 25 - > VIS) is Ils 247 47 Os Two D Toutes volt q WF I Stept. Applying nodal analysis at VLB) us Vis) (s(s+H) +H) - 10 (Sth) VIS) [5+45 +H) = 10 = V(s) = ) V (s) = 10154H) St4u - 10 (8 +45+4)

ves) steps: Is) st4 lolstH) 315445+4) {3+4) S15+48ty 4. IIS step Vols) = = 4 x 10 s(s+HS+H) Sist step Applying Applying inverse laplace transtorm. volt) - 10 ult) – et ult) + ate at ult) a loult) (1-ezt tateet)