Question

Condsider the ODE d2 1 (t) + 50 x (t) = F(t) dt2 where the forcing function is given by the Fourier series F(+) = 0 +21 on sin (nt) with co = 9, c1 = 10, ... Assuming a particular solution of the form Ip (t) = a0 + Anal (an cos (n t) + bn sin (nt)) find and enter the exact values of an and bn requested below. 20 41 == 61 - 10

Answer 1

2 We have da(t) + 50 xlt) = F ). dt? where F(t) = Cot I cn sinnt) nal Now, the particulars Solution Xp (t) is. of the form Xp (t) = at ² (an Cosnt t. bn sinnt ار nt) nal Smees Xp (t) is a particular solution, equation 0. So, it satisfies Then, d. Xp (t dt E Fay xn) Sin(nt) + = (bnxn) Cos ont n=1 n=1 d xp (4) Ż Eanxen?) Cos (nt) + £ -tbnxn) Singht) at nal nal So, putting this in equation , we get. Ž Ganxst) Cosínt) + Ž G-boxny) Sincent) nal nal + 5000 + (50 an) Cosen t) + 3 (50bn) sin(nt) nal nal = CotE Cn Sin (nt) n=1

which gives. 50 at an (50-73) Cosnt + 2 bn (50-12) sin(nt) nal nal ina) of Cosent), = Cot Z en Sinnt.) Comparing from both sides, the co-efficients sin (nt) and constant terms. we get 50 ao=co an (50-22) = 0 bn (50-27=cn Qn=0 vnxi [: »'+50] and bn (50-1²)= Cn 9 ao = Co 50 9 Vn7 Co=9,0,-10 Now, we are given that So, ao=co 9 50 50 a=0 [: an=0 vnxi] and bi Ci 50-10? CI 49 => bi Cla 10 49 Hence, ao = 9/50 a 0 b, li 10749

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