Question

a. Find the most general real-valued solution to the linear system of differential equations x = -[42]; xid) + c2 x?(༧) 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

Answer 1

a. @ Find the most genral real-Valled solution to the linear systers of differntial equations to 4 2777 A 4 잇 -3 First Find eigeny alles 1-11] g Fl-4 3 1-1 Find the determinant of the given metrin 2 12-1 E -1-1 q 1 +31 42 1-1 This is Characteristic Polynomisel. solve the equation la+30+2=0 The roots are de -9 7 . -1 دار و Nent Find the elgenvectors. @d=-1 - 1-4 2 kr T B-1 ] -3 ng Preform 1-1 opration to row Rg → Rg - Ri Roth Ri L CS Scanned with CamScanner

Now, Solve the matrin equation V, D ] O Vg If we take Vook, then k, then I, V.- 2*, Vsak 3 qk Therefore va 3 k o 4 =-g .g 2 R3-g2 R3 - 34 Penn i. -3 RI Ri +1 o] Now, solve the equation 11 Vi ID) 61 If we talke Vg zk =) Vi=k, Ug = k Therfor VEK k k CS Scanned with CamScanner

=) elgenvalle dil, eigenvector 913 and g -g eigen vector 1 bunrral Reel- Valled Solletion 130) - cieply?46:24 | % C +Cg et Ansb. bath eigen Value Real, unequal and negative System sink / stable made. Consider the linear system -6 اع -9 15 ។ Find the exenvells and espervectors for the Coefficiend metrin Ang : AD -9 a I 15 9 For cigon Valle 6-19] CS Scanned with CamScanner

-d-g 15 g-/ Find the determinant of the obtained matrin -2-9 -6 1 + 9 15 g-1 Solve the equation 1+9=0 & The Roots au λι -Bi dg = 3; Next Find the eigen vectors. 1=-3 (-9t3 P-1-9 -6 -6 15 15 9-11 970- add row 1 multiplied by 3 tc to Row a ( Ro > Rgt + 134. TRI) Tag +3 و ا م ( - DOS Now, Selve the matrin equation 3ti 5 2 + 1 CS Scanned with CamScanner

If we take Vo = * then Vis K(3+0) V2 =k 5 K(370) 2 Therefore va 5 k 6 dani Tg-Bi 1-gozi -6 -6 Rg Rg + (1 - 1 -15 g-Bi RI >I Rg g-3i 1 Mlco Yy d Now, solve the matrin equation -& inly 1-6] d Vg zk 5 If we take Ug = k, then Vi= k1-3 + i) * (-3+0) -2 / tis K Thertore [*** 1,=-3i, V - 2 3 - 1 2 + 3 and olg = 31, Vo = CS Scanned with CamScanner