Question

Use Laplace transformation to solve differential equation.

*+ 4y = e', y(0) = druge dt - dºg(0)

Answer 1

given differential equation dy +4832 0(0)-decod 9600 i. yall +44 = et, y(o)= y CO )= y "(0)=0 applying Laplace transform on both sides L { y'"(t) +49 (H)] = 2[et] ne We know that L[y" (t) ] = 83 [ych]- 3 y co) - 8 Y'co) - y"co) Leat] =ta h[y"(t)] #42[y(+)] = [es 8° LC9CH)] -82400) - SYCO) - y"CO) + 44590+)]== 8 L[YCH)] -8°(0) -860) -(0) +4 LC96+)3= , (8²+4) LC y Ct)] = 1 / 4 Lcgct?] = 8 (44) use partial fractions (S-1) C+4) = put s=1 = A (844) + (BS+ c) (S-1) 1 = A(1+4) +(B + c) Col. 1-5A A=I

comparing 82 coefficients 0 = A + B = 3 = -A B = -2 companing s coefficients 0 - - 13+ =) c= 13- S-1 Ict) u- - [3Ct) 3 = 16 ] YC [ ] C- 13) we know that 2 [a] = eat 1 2 3 2 7 = cosat TC ] a sinat Y(t) = 3 [et - cosat - sinet] Y(t) = - et - - cosat -1 sinat TL Ta