Question

Question 3

1. Sol orms to solve the following differential equations: 11se Laplace transforms to 47'(t) + 4y(t) = 9e' with y(0) = y(0) = 0. x) = 5 sin 2t with y(0) = 0 and y'(0) = 1. 2. y"(t) + 4y'o 3. y" (t) – 4v

Answer 1

yh(E) - 4 yle) = 5 Sinz+]; 703=0; y(0) =1 Concept of Laplace Transforms of desivatives: - Låg (t)} = SY(s)-yo) where 46)=Lqy(t)} 4f4"b)} = 5:46)-40) - yle) -$4"[)= 82408)-546) – sy 60) – y") 49 993)] = s4y69)- s’yo) -s’ylo) -sy'to) - y"C) Now to solve o first apply Laplace transforms on both sides to Ly9|)3-44{41}} = 54{size} Note:- Linearity property of Laplace Transforms:- La flt) + b glt) + Chlt)} = AL{fe Y = b { glty & CLşhit} > (346) - syo) - yle)- 4469 = 5 Note: - Ls binaty = a → 846) -s() -1 -440) - Liswa ya] = 524) ) = + | 1-0)- Tony stay to

Now after getting YG) Apply inverse Laplace Transforms on both sides - L {y(s)} = a L (19} -ao { colocopter {3} to tima u { retuscan ] 따 method s²tysty va = As + B + esto (using partial Fractions) (574) (54) (ASTB) (54) + (50) 544) 845 (544) (541) => [CAs+ b)(+4)+ (Cs+ p) (S?+4) = 1] >> As3_49) 5+B5?_4b+c83+40)5+ 05740-1 - (A+C) 5 +(B+D) s t u CC-A)S + 4CD-B) = 1 How comparing coefficients of 33, 5is, constant terms on both sides we get (A+C) =0 |(B+D)=0 14CCA)=0| 4CD-B) = (A=-c) =B=-01 C-A=0 D-B=1/41 (i), (T) = A= Col (i),(N) => D-CD) ly =) 20[D=112-0) (V), (i) B=-1/2 1

16 (Ehes con esta en 4 (ar th] => ["{ ys} over { flow czas} + { stor} A L' {y00% = 1029 {}[ * fr]} <L"{4} 9 29 44033 = 5v { da taj ty Pisto} akt {1073=530433-c19.1? +214 } sfor? vosita's Point 122.+0? ) To find 21 { 1 / 4 3 - y (5+2) CS-2) St2 s.2 → ACS-2) + BCS+2) = 1 put s=-2 A (-4)=1 = A=-14 put S=2 B(4)= 1 B=114 = 1/2 = (552 - st) Lutfufty-{s}® a

o[ { 4833=5{ $(etc+) - 4 sinet] +4(2***] * LT 2 2 = Sinat L = [1 { staz Y = ta sinat = [1 { 1 + 2 } = 21 { 1 } = 1 Sinity + L {46094 = (44) – { sim 2t + t (eltet) L {46)} = 6 (e?tet) - s sint » Lt {y(s)} = 3 (et e 24) - s finzt iy) - 3684_e_) -(5 Siner) | 2 fi ef sce)= 408) = +{X593= yey]