Question

Differential Equation

Please answer both of the questions below Thanks!

Solve the given initial value problem. y'' + 36y = 0; y(0) = 3, y'(O) = 5 x(t) =

Answer 1

= o y +36y=0 Y"+36y=0 y 10) = 4'10) 25 Ausiliary Equation m²+36=0 m² = -36 m= +61 solution will be yt) = C, coset & Casinot given yo) = 3 3 GX1+exO (=3 y (0) = 5 5 = GX-6sinet + 6x6o6t t=0 5 GX-6X0 +27641 G = { Then exneral solution will be yt) = 3 og 67 + 5 sin but

It y = pot Amiliary Can let y"+ 2ylty = xit m² tomt 1 - 0 (m+ = m=- -| Saltion will be complementry function, C.F. = (atletet = Gott Gutet for finding P. I. tact h = s55 et Y = 7 et + etxzot at Then u + C y get) at JY, Y - Y YI [- tet tet tetet (+2 est at u= (+2628 dt - test test teet at y = (+2dt - t2t

and u= (- 4 gt) at yiya - Ye Yi - tot eetat et [- tette t] + te txost (tzt dt - text, p2t + toat v = 1 = Auto = 5-34 (-atest at -2 tdt u=-4² Then P.J. = vyit uy = -t yet2txtat = -t et tot & t - tect YH) = C.F. + P. I Then Genesal solution will be y(t) = Get toute t + [ttet ] yht) = [+ Cet + f2J et