10. Find the general solution of the system of differential equations d 3 -2 -2 -3 -2 -6 dt 10 2 2 1 + 2te“ + 3tet + 4t°e* -3 Hint: The characteristic polynomial of the coefficient matrix is -(A– 4)²(A – 3). Moreover 2 1 Xp(t) = t²e" + t®e4t +t*e3t -1 -3 is a particular solution of the system.

Theorem 1. The following system of equations Un-1Vn-3 Un-3Vn-1 Un+1 = and vn+1 = (35) Vn-1(A+ Bun-1Vn–3) Un-1(C+ Dun-3Vn–1)' (1-А) has a 2-periodic solution if u_3 = u_1, v_3 = v_1, u_3v_3 = u_2v_2 = " and A = 42 C, B = D+ 0. 41 -A), where A = C,B = Proof. Let u _3 = u_1,v_3 = v_1 and u_3v_3 = u_2v_2 = D in the exact equation (34a). Then 2m-1 2m + (1-C) Σ C n-1 l=0 U4n-2 =u-2 |T 2m A2m+1 + (1– A) E A! m=0 1=0 =u-2. (36)

matrix algebra qusetion need help solving it.

Consider the second-order system of differential equations dy d²x dt² =x-5y, The coefficient matrix 1 -5 eigenvalues are 6 and -4. Select the option that gives the general solution of this system. I dt² C₁ -5x + y. 1 -5 11 1 has eigenvectors Select one: [*] = C₁ [ ¹₁ ] cos(√õt) + C₂ [ ¹₁ ] sin(√6t) + Cs [1] cos(2t) + C₁ [1] C3 C4 sin(2t) evot + Vot C3 - [3] = a₁ [¹₁] ev [*] = C₁ [1] cos(√ēt) + C₂ H ₂[1] HHHHH [+] = C₁₂ [ ¹₁ ] ev√² + C₂ [ ¹₁ ] e-√₂ 3 [1] 2²¹ -C₂₁ [ ²₁ ] e-√² + C₂ [1] sin(√6t) + C3 [1₁] [1] Cos evot +03 e2t+CA C₂ [4] [1] and H cos(√6t) + C₂ and the corresponding cos(2t) + C4 H e2t + C4 sin(2t) -2t [¹1] e ²2 -2t e2t sin (√6t) + C3 as He" + a₁₂H₁² -2t

Consider a system of differential equations which has the general solution: 18]- = C, · cos (5. t) – sin (5 - t) )+C2 · sin (5 - t) + • Cos (5 - t (a) Determine the fundamental matrix (t) which has the property (0) = I. (t) = (b) Use the fundamental matrix found in part (a) to quickly determine the particular solution for the following initial conditions: r (t) : y (t) = :8]-[- (0): 1 (ii) r (t) = y (t) =

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linear algebra

Question 1 of 2 d.x dt (11+x)(y − x), = (3 − x)(y + x). Find the corresponding linear system, J, near each critical point, if possible. Find the eigenvalues of each linear system. NOTE: Enter the exact answers. - / 1 ⠀

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