utions of the differential + br + c = 0 and that satisfies (y2/y1)' = W[y₁, y21/y, where W[y₁, y2] is the Wronskian of y₁ and y2₂. Then use Abel's formula (equation (23) of Section 3.2) to determine y2. 711 1 Then think of r₁ as fixed, fo(t; ri, r2) as r2 → r1, case of equal roots. ri, show that = a(r-r₁)²ert. (37) 7) is zero when r = r₁, it -]=ay" +by'+cy = 0. ect to r, and interchange with respect to t, thus tet] 2ae¹¹ (r-r₁). (38) 6) is zero when r = r₁, of L[y] = 0. method of reduction of fferential equation. = 1² = t = t-1 = sin(x²) Y₁(x) = x-¹/2 sin x uniform stream past a 5x2/2) is one solution, of an integral. tended to second-order a known nonvanishing hat a second solution y2 In each of Problems 25 through 27, use the method of Problem 24 to find a second independent solution of the given equation. Y₁(t) = t-1 26. 25. t2y" + 3ty' + y = 0, t> 0; ty" - y' +4t³y = 0, t> 0; 27. x²y"+xy' + (x² -0.25) y = 0, y₁ (t) = sin(t²) x > 0; y₁(x) = x-1/2 sinx Behavior of Solutions as →∞o. Problems 28 through 30 are concerned with the behavior of solutions as t → ∞. 28. If a, b, and c are positive constants, show that all solutions of ay" +by' + cy = 0 approach zero as t→∞. 29. a. If a > 0 and c> 0, but b 0, show that the result of Problem 28 is no longer true, but that all solutions are bounded 30. as t∞. b. If a 0 and b > 0, but c = 0, show that the result of Problem 28 is no longer true, but that all solutions approach a constant that depends on the initial conditions as too. Determine this constant for the initial conditions y(0) = yo, y'(0) = yo - Show that y = sint is a solution of y" + (k sin² t) y' + (1 - k cost sin t) y = 0 for any value of the constant k. If 0 0 and k sin² t≥ 0. Thus observe that even though the coefficients of this variable-coefficient differential equation are nonnegative (and the coefficient of y' is zero only at the points 0, π, 2π, ...), it has a solution that does not approach zero as to. Compare this situation with the result of Problem 28. Thus we observe a not unusual situation in the study of differential equations: equations that are apparently very similar can have quite different properties. t = Euler Equations. In each of Problems 31 through 34, use the substitution introduced in Problem 25 in Section 3.3 to solve the given differential equation. t> 0 31. t²y" - 3ty' + 4y = 0, t> 0 32. ty" +2ty' +0.25y = 0, 33. t²y" + 3ty' + y = 0, t> 0 34. 4t2y" - 8ty' +9y = 0, t> 0