Question

Find the Laplace transform of the given function

f(t) = { 0 < t < 2 t 22

Solve the integral equation

Answer 1

Sol: 61t) - $t t o < t < 2 t > 2 2 2 We know 0 L {6 (H)} = gerst f(t) at - 1 ***)33 34** *48144 Jestfundi - Jest et dt + Jest. 2dt 2 dt 2 2 2 Jest tat- tat-festa tedt +. Jzest [[+]+ Jest (2-t)dt Lff (t)} = L[t] + I L[t] + Is (let) Where, I = { erst (2-7) at Ig , we put s 2 For Ig (+-2) - V + t-V+2 at t=a + V = 0 at t=0 => V = ∞ differentiating. (t-2) =V we have at = dv 1

i. Is S (172) 42-(4+2)} dv ro -SV-25 e (-u) dv Is 12.10.00.1** (vid I; --2-** Seev cudo 18 (^-) -25 FI 11 ce-25 Jest | 64) dx = pit) at (t)dt In seas L[+] ..L{t(+)} = [[t] + (-2+25L [+]) = 1 [+] - (-25 L [t] = L [t] (i-pero e-25) . 82 1 eos Se es = $(7) 4$7!, e 25 1 2 25 S e (2)

t 2 s(t) = 4+ - 3 6 y(z)sin (+-x)dz t o "We know that, An integral equation of the forem Y (t) = f(t) + F(t) + Sk (t-u) Y(u) du is called integreal equation of convolution type. And The above equation is also expressible as Y(+) = f(t) + K (+)*Y(+) where, * represent multiplication. 2 The equation can be expreessible as y(t) = 4+ - 3 Sint * y lt) Taking Laplace transform we have , Lf y(t)} = 1845} - L{3 sint * y(t)} » L{4 (+1}=46&t} - 8L{sint} * Lfylt)} P-L846+)} > y(s) = 4 . 2 / 2 3. 1 si+12 y(s) = y(s) 4 3 y(s) 5²71 => y(s) - 82 * y(s) + 3 52+1 4 y(s) Se 3 4 => y (5)(1+ ) 5 ²+1 S2 3

. २५6) (52+4+3 4 52 62+4 4 ५(6) (54) S+4 S+4 8 y(5) 82 ५(6) . S2 4 (+) (544) 4 (3+1) (S+4) 3 5२ (544 4 |२. 5२ (5२+4) ५० (6) 5 sa Taking in verese laplace treansform. * :-'६५.ts)3 = 18,43 - 1157) EL{yt} ५ (t) = 41-19 -1२ 15ो S२ (२+4) = ५८७) १५) LL-4yo १८६ 4 (s+4 । 1 131 - 12 २ S 452 रे ५(t) = 41 ५(t)- 41-4 41-5) - R*4:१६२१+१२४ ५ (t) - 4* * - 3xt + 3 x ** गे S 1544 J 15२+ २२ 0

=> ylt) 4t-zt + 2 * sin at a => yt) w t sinat 2 -". The required solution is. y (t)= t + 3 sin at 2 UL * Feel freee to ask if you have any doubt regarding the solution. Please upvote (like) if you understood the solution 6