Question

(1 point) Consider the linear system 3 a. Find the eigenvalues and eigenvectors for the coefficient matrix 0 and A b. Find the real valued solution to the initial value problem -392 5y + 3y (0) 9, y(0) - -10. Use t as the independent variable in your answers, (t)

Answer 1

3 А. 3 3 -2 5 3 Now Characteristic ograsion of A is -3- 3-7 - (3+x)(8 - x) +1020 -g+x²+ 1000 12 tiso n ro so eisen value is ti. eisen nector conosponding to eigen tee value C -2 24 3-i 2 0 are same Sa+B-ja, so and (-3-1)2 - 2x2 = 0 2= (-34972 [stree [since this two quasion alt x = x 3+ ya 1.341 Where KER) Then ད ། SO, ng 1 So, eisen vector Whem ka 5 1 * K=)) croreponding eigen value aiai. Now , we find eisem veetor conosponding to eigen 22 = -2, alt eisem talul vector Nalue be 92

anen, AY = 727 04 3t' and oo. (-3+jxy, - 24, 20 54, +(3+ºj 920 equasion 4,5 - 3532 so. Smee these two are some 50 det =s S, So 42 92 (wherre SER) 1 be the eisen vector (when s=1) conosponding to eigen rame to eisem name 72=-0. -3+0 so, Mozū

Here the inisial value problem is y = -34, -292 = 59, +342 (0) - 9 92 (0)=-10 So This can Go written as -3 2)5, 3(0) = (-70) 3 Where NOW, ligem value aile til From port a a and lisem vector are 1 det Po simee lisen vector con be written and inem 3) B we choose ara Perin for p P AP 3 = Who 3 - 2 5 M Cost Sint ct e e Simt Cost

Simce of co B from them em this の B Cos ßt ( cosßt et dt Simßt e e Simpt therefore the solusion of the system () is ct It = pe o Yo Ś cost - Sint sint cost BIS OG COSE -Simt 12 300 sint 15 cost + 10 Sint 12 15 Sint - 10 erst 15 cost +10 Sint) - ( 15 simt -lo cost 1) 15 sint 10 east (Beast + 2 sint a simt to cost 15 sint - 10 cost ra cost 7 simt 15 sint 18 east 9,6t) : I cost-fsimt I (t) = 15 simt - 10 cost