Question

Transform the given initial value problem into an algebraic equation for Y = L{y} in the s-domain. (a) /'"-6y" +1ly - 6y=et, y(0) = '0) = Y(0) = 0 (b) y" + 1" + y + y = 0, y(0) = 1, y(0) = 0, y"0) = -2

Differential Equations

Answer 1

Soln cas consider the differential equation y"-byl tllyl-6y=ele Take haplace transfoom on both sides, wegel LEY'l-oy" tiyl-6y]= [out] = (53y - spyco-sylco) - y"0))-6(say - syron -yicos) +1(sy - You) – 67 = tu using your y con= yilcev=o, we gel sBy – 6534 Hisy -67 = stik >> (83-65° +115 -6) Y = sta sy = (3-4) (-63* Hlls-0) so the required transform function is so domain is Y = LSở } - - (S-4) (83-683+15-6) CANS) (b) consider the equation y "tyll tyl ty=0 Take Laplace transfoom on both sides, we get equation

L[y'll tylltylty]= 0 >> (siy - søy (o) - ylco.s-ylico))+(52 y = 5 y(0) – y lo)) : +csy- y)) + y = 0 =) (337-52-0 +2) + CSPY-5-0) + (SY-1) +y = 0 as yro) = 1, YICO)=0, y CO=2. on simplifying, we get (33y +say ts y+V) -s? +2 -5 -1 = 0 => (53 +52 +5+ y = s?ts1 3) Y = sits- . 3+si+st! so the required transform functions Y=2193= sitsal 53 +5² +st (ans)