Question

x1 x2 +10 cos t x1 3x1 x2 10 sin t Find a solution for the given linear ODE system given above using the method of undetermined coefficients

USING UNDETERMINED COEFFICIENTS!!!

Answer 1

lo Coy? -l0 Sint 3 -1 we sisnst ind the Salutron od the homogeneous System. Calculata the eigen values of A o >-1-A) (+)-3 = 0 ?24=0 ?=?2 A-SIIO 1- I r-1- E 99 T detormine the eigen vectosr vi(Vil V21) fasr the falua i 2 VII (A-I)V O i 3 -3 R2-7 R2+3R [-1 VII 1 V21 Vn tV21 = 0 wa choose V2=1. (1,1) Far Sim alicity So Vin =1. Thesrafogree vi Shomilarly 2=(V12 V22), Cosorespo nding to tha eigenvalua 2 (A-A2?) V2=0 11 we ind the eigen vectosr \r\nVI2 V22 3 R2-7 R2-RI 1 V1 2 V22 3 V12 t V22 = 0 VI2 V22 Page-2 dor Simplicity we choose 22=3 So, V12=1Thesrefogre V2=C1,3)T Thus the goneral solution of the homogoneous system 1s 2t 2t Q. Xolt)-Tt) =92 pasticulasr Saltron wo Sind Sorm similar to dlt) ia in a assume that Xt= Gst +Ci7Sint di bl lt)= a, Cost +GSint 2lt)= b1 Cast +d, Sint \r\nGt - Sint +C, Cost xt b Sint + AStor the Substituhon in the osriginal Page-3 di Cost , we obtain. a,Sint +ciast = +b a34 4d, Sint t1o Cost 9,Cost +G blittd ast= 39,at +3C, Sint bICost-d, Sint-loSint L4uating the coesticionts os the tenms -b,= 3 C1-di-lo-5 -a1=C, +di (3 3a,bi d, aC, be comeS. 3C1 +a,+-l0 a1+4C-10 -a-C=391-b becomes -a,-C-3a = -b1 491-C -b a,t 401-10 -b -C + 4 + 591+50-lo O a1+C= 2 \r\nbecomes -91-0,=2 becomes 3a1-b1-2 $beeonaes aeaGA24LO -bMeed 8 3artr2Cp8 2 eAfaCA 2+8-6 Paga-y 4) becomes 2-a,=a1 + b,+ 10 a,- bi= lo-2 a 2a-b 8 0 becomes 341- = -2 2a, , = 8 591=-10 => a 2-9 - 6-b,-2 - -br =4 b +47 Sint Cost Xl?) = -2. 2t GttSnt Ans "",""error"":null,""modified"":1560267406,""source"":""automated""}"