(a) Find the general solution for the following linear homogeneous recurrence relation an = 2an−1 + 15αn-2, α0 = 7,α₁ = 25
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- 4. Let F(n) = 2F(n – 1) + 5F(n – 2) – 6F(n – 3) with initial conditions F(0) = 4, F(1) = -4 and F(2) = 2. (a) Solve the recurrence equation using the characteristic equation method. (b) Solve the recurrence equation using the generating function method.Given the differential equation: Choose all correct answers. A (x²-1) y" - 6x y¹ + 12y=0 D The recurrence relation is given by: (n-2)(n-3) C C n+2 (n+2) (n+3) n y=c (1+6x4+x³) +c₂(x+x² + x³) (C) y=c (1+6x²+x4) +c₁(x+x³) = The recurrence relation is given by: C n+2 = (n-3)(n-4) (n+1) (n + 2) C n n = 0,1,2,... n=0,1,2,...Find the general solutions of DE
- (a) Given that (x – 1)³ (x – 2)(x - 3)(x – 4)² = 0 is the characteristic equation of a linear ho- mogeneous recurrence relation with constant coefficients. Write down the general solution An of the recurrence relation. (b) Solve the initial value problem of the linear homogeneous recurrence relation with constant coefficients an = 2an-1 - an-2 with ao = 1, a1 = –1.Solve the linear recurrence relation (find a formula for x k in terms of k ): Xx+2 = 2xx – Xk+1 Where x o = 1 and x 1= 2. You must use the eigen values to get the credit.Consider the following linear system of differential equations. Y - 324y2 Which of the following is the general solution to this system? e108x, -108x (C) c1 e108 -108x (A) c1 (B) c1 +c2 e$4x. -54x (F)c e108r (D) c (E) c1 -108x (G) c1 e108x -108x (H) c1 -54x + C2
- 20 (a) Check that the solutions to Ax = 0 are perpendicular to the rows of A: 4 2 0 1 [100] 1 0 0013 = ER. 0000 3 4 1 (b) How many independent solutions to ATy=0? Why does yT= row 3 of E-¹? A 2 =The technique works for equations of any order, for example, first order. If is a solution of the differential equation ak+2 then its coefficients a satisfy the following recurrence relation. For k ≥ 0, We also have 2 k+2 2 48 a₁ = -2 ao. help (numbers) Therefore, write the first few terms of a solution where y(0) = 2 y = 2 ak+1+ 4. 1 k+2 +-4 x² +-2 + help (numbers) x+ y=[akak k=0 2³+ y + (2x + 2)y=0, a. help (formulas)Suppose that each year, 80% of residents living in state A stay in state A, while 20% of them move to state B. Also, 70% of the residents living in state B stay in state B, while 30% of them move to state A. Further, assume at t = 0 years, there are 2 million residents living in state A and 5 million residents living in state B, and let xn and yYn represent the populations (in millions) of the states A and B, respectively, at time t = n years. Given that (:) - (: )() Xn a b Yn d 5 find the value of the number h.
- If d³p dq³ 2 + d²₂ ²p dq² 3 + p = 0, what would be the order of each differential term?Consider the initial value problem - [] 3 *+ f (t), *(1) = Suppose we know that -2t + a X(t) = 212 + B is the unique solution to this initial value problem. Find f (t) and the constants a and ß. a = help (numbers) B = help (numbers) help (formulas) help (matrices)Q12\ Solve the equations: 1-j (1) e3z ; (2) In(z+ j)= jt/6 ; (3) z* = 4+ j3 1+j