homogeneous solution for n>-0, a) y[n] + 1.3y(n-1] + 0.4y[n-2] 0; y[-1-0, y[-2] 5 yn]- 1
3 Q 3 homogeneous y(x -y) =0, z>0
5) Consider the second order linear non-homogeneous differential equation tay" - 2y = 3t2 - 1,t> 0. a) Verify that y(t) = t- and y(t) = t-1 satisfy the associated homogeneous equation tay" - 2y = 0. (5 points) b) Find a particular solution to the non-homogeneous differential equation. (10 points) c) Find the general solution to the non-homogeneous differential equation. (5 points)
Q #1 (2 marks) Find the general solution for the DExy'-ymx1y, x>0, y>0.
5. Repeat the same questions in 4.) for the ODE Py"- tt+2)y+(t+2)y2t3, (t>0) (a) Find the general solution of the homogeneous ODE y"- 5y +6y 0. Particularly find yi and (b) Find the equivalent nonhomogeneous system of first order with the chan of variable y (c) Show that (nvand 2( re solutions of the homogeneous system of ODEs (d) Find the variation of parameters equations that have to be satisfic 1 for y(t) vi(t)u(t) + (e) Find the variation of...
3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a value ofr for which y = et is a solution to the corresponding homogeneous differential equation. (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation. (c) Use Variation of Parameters to find a particular solution to the nonhomogeneous differ- ential equation and then give the general solution to the differential equation.
Let (In), and (yn).m-1 be sequences such that Pr – yn| < 1/n for all n. Use the definition of convergence to prove that, if (2n)_1 is convergent, then (Yn)-1 is convergent.
5. Given y,-t is a solution of t2y" + 2t1-2y = 0, for t > 0, find a second solution y2.
Suppose that the population has the following pdf: Le-(y-0) if y> 0 f(y) = { 0 otherwise Let U1 = min{Y1, ... ,Yn} and U2 = $1=1 Yį. (a) Show that the pdf of Uı is f(y) = ne-n(y=0)1(y > 0) (b) Show that U1 - 1/n and U2/n - 1 are both unbiased estimators of 0. (c) Find the variance of each of the unbaised estimators in part (b). (d) One of U1,U2 is a sufficient statistic. Which one?...
(1 point) Find the solution of xy" + 5xy' + (4 + 1x)y = 0, x > 0 of the form Yi = x n=0 where co = 1. Enter r = -2 Cn = - n= 1,2,3,...
7. (10 points) Find a particular solution yp(t) to the nonhomogeneous equation ty + y - y = 24t*, t> 0, given the fact that the general solution of the associated homogeneous equation is yn(t) = cit + cat-, C1, C2 E R