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Homework Help
Calculus and Beyond Homework Help
Solving a system of diffy q's with complex eigenvalues
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[QUOTE="Jamin2112, post: 2675021, member: 222266"] [h2]Homework Statement [/h2] Express the general solutoins of the system of equations in terms of real-valued functions. [B]x[/B]'= [1 0 0; 2 1 -2; 3 2 1][B]x[/B] (I wrote the matrix MATLAB-style) [h2]Homework Equations[/h2] The coolest equation ever: e[SUP][I]i[/I]b[/SUP]=cosb + [I]i[/I]sinb [h2]The Attempt at a Solution[/h2] Assume [B]x[/B]=[B][U]R[/U][/B]e[SUP]rt[/SUP] (the underlined r is an eigenvector) Determinate[1-r 0 0; 2 1-r -2; 3 2 1-r]=0 --> r = 1, 1+2[I]i[/I], 1-2[I]i[/I] r = 1 --> [0 0 0; 2 1 -2; 3 2 0]([B][U]r[SUB]1[/SUB][/U][/B] [B][U]r[SUB]2[/SUB][/U][/B][B][U] r[SUB]3[/SUB][/U][/B])[SUP]T[/SUP]=(0 0 0)[SUP]T[/SUP] ---> [U][B]R[SUP]1[/SUP][/B][/U]= (2 -3 2)[SUP]T[/SUP] I do the same with the other eigenvalues, and come up with 3 eigenvectors: [U][B]R[SUP]1[/SUP][/B][/U]= (2 -3 2)[SUP]T[/SUP], [U][B]R[SUP]2[/SUP][/B][/U]= (0 1 -[I]i[/I])[SUP]T[/SUP], [U][B]R[SUP]3[/SUP][/B][/U]= (0 1 [I]i[/I])[SUP]T[/SUP]. By Superpsition, the full solution will be [B]x[/B](t)=C[SUB]1[/SUB]e[SUP]t[/SUP](2 -3 2)[SUP]T[/SUP] + C[SUB]2[/SUB]e[SUP]t[/SUP][SUP]e2[I]i[/I]t[/SUP](0 1 -i)[SUP]T[/SUP] + C[SUB]3[/SUB]e[SUP]t[/SUP]e[SUP]-2[I]i[/I]t[/SUP](0 1 i)[SUP]T[/SUP] = e[SUP]t[/SUP] [ C[SUB]1[/SUB](2 -3 2)[SUP]T[/SUP] + C[SUB]2[/SUB](cos(2t)+[I]i[/I]sin(2t))(0 1 -[I]i[/I])[SUP]T[/SUP] + C[SUB]3[/SUB](cos(-2t)+[I]i[/I]sin(-2t))(0 1 [I]i[/I])[SUP]T[/SUP] ] ................................. This somehow simplifies to the answer in the back of the book, C[SUB]1[/SUB]e[SUP]t[/SUP](2 -3 2)[SUP]T[/SUP] + C[SUB]2[/SUB]e[SUP]t[/SUP](0 cos2t sin2t)[SUP]T[/SUP] + C[SUB]3[/SUB]e[SUP]t[/SUP](0 sin2t -cos2t)[SUP]T[/SUP]. I don't understand the simplification process. Yes, I know the imaginary numbers just get absorbed into the constants; but I can't figure out the rest. [/QUOTE]
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Solving a system of diffy q's with complex eigenvalues
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