Question

(1 point) a. Find the most general real-valued solution to the linear system of differential equations.' = r. -6 x1 (1) + C2 x2 (1) b. In the phase plane, this system is best described as a source / unstable node sink / stable node saddle center point / ellipses spiral source spiral sink none of these

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Answer 1

Let Then * (8-451-o zives 12-* ) x² + 2x tun +8+150 x2 +60 +9 20 (a +32 20 -3, -3 for 2-3 (B-2I) x ao gives x+y= B=(-) is the eigen rector corresponding to 2=-3 The solution is -3€ 7 = ci est le cu tt) where w find the way B = -3 m ti (B+3I) W = (0:03

This implies P + P2 = 1 put P = e. then P2 = 1-e w = ( ) = (-e) = + (!). 2003 () + 6s "{() ++E)+(-)} -0,3"(!) + 0,6"}()*+(..) Hence -34 -3t n, (t) Halt) ] = C, C2 Do- -34 .37 - te

[3] Let A= 8 -6 S Then -9-2 8 (A xII to gives -6 5-x OY -45 +22+90-5x + 48=0 OY x2+un+3 =0 or (x+3)(x + 1) = 0 x = -3 -1 3 are eigen values of A. Now for 2= -3 (A-2I) x = 0 gives نما - 6x + 840 - 6x + 870 1.-3 is the eigen vector corresponding to a=-3 Again for = -1, (A-21)x=0 gives - 8x + 8x = 0 (!) is the eigen vector corresponding to 2= -1 the solution is - t X(t) = е, е [3] + € * D'] - 36 40 e n, (t) 14 e 3e -34 1₂ (4) real and distinet and Here also same two ergen values are sign with negative. Sink / stable node Hence