Recitation 23 There is a double-root at 1, so the homogenous solution has the form a+ Theres an inhomogenous term, so we also need to find a particular solution. Since this term is a constant, we should try a particular solution of the form Xn=c and then try Xn=c+dn and then Xn=c+dn +en and so forth. As it turns out, the first two possibilities don,'t work, but the third does. Substituting in this guess gives 2Xn-Xn-I c+dn+1)+e(n+1)2=2(c+dn+en2)-(c+d(n-1)+e(n-1)2) All the c and d terms cancel, so Xn=c+dn-n2 is a particular solution for all c and d. For simplicity, lets take c=d=0. Thus, our particular solution is Xn=-n Adding the homogenous and particular solutions gives the general form of the so- on Xn=a+bn -n' Substituting in the boundary conditions Xo=0 and Xw=0 gives two linear equa tions 0=a+b The solution to this system is a=0 and b= w. Therefore the solution to the recurrence equation is Xn=wn-n=n(w-n (c) Return to the original problem, where Stencil has the Cliff of Doom 1 inch to his left and an infinite plateau to this right: What is his expected lifespan there? Solution. In this case, his expected lifespan is lim1(u-1)=∞ Yes, Stencil is expected to live forever! (d)Compare your answer to the previous part and your answer to the last part of the previous problem. Anything troublesome? Solution. So, Stencil is certain to eventually fall off the cliff into the sea-but his expected lifespan is infinite! This sounds almost like a contradiction, but both answers are correctRecitation 23 5 There is a double­root at 1, so the homogenous solution has the form: Xn = a + bn There’s an inhomogenous term, so we also need to find a particular solution. Since this term is a constant, we should try a particular solution of the form Xn = c and then try Xn = c+dn and then Xn = c+dn +en2 and so forth. As it turns out, the first two possibilities don’t work, but the third does. Substituting in this guess gives: Xn+1 = 2Xn − Xn−1 − 2 2 2 c + d(n + 1) + e(n + 1)2 = 2(c + dn + en ) − (c + d(n − 1) + e(n − 1) ) − 2 e = −1 All the c and d terms cancel, so Xn = c + dn − n2 is a particular solution for all c and 2 d. For simplicity, let’s take c = d = 0. Thus, our particular solution is Xn = −n . Adding the homogenous and particular solutions gives the general form of the so￾lution: Xn = a + bn − n 2 Substituting in the boundary conditions X0 = 0 and Xw = 0 gives two linear equa￾tions: 0 = a 0 = a + bw − w 2 The solution to this system is a = 0 and b = w. Therefore, the solution to the recurrence equation is: Xn = wn − n 2 = n(w − n) (c) Return to the original problem, where Stencil has the Cliff of Doom 1 inch to his left and an infinite plateau to this right: What is his expected lifespan there? Solution. In this case, his expected lifespan is: lim 1(w − 1) = ∞ w→∞ Yes, Stencil is expected to live forever! (d) Compare your answer to the previous part and your answer to the last part of the previous problem. Anything troublesome? Solution. So, Stencil is certain to eventually fall off the cliff into the sea— but his expected lifespan is infinite! This sounds almost like a contradiction, but both answers are correct!