Problem 1 Consider the Euler equation t2 x00 + αtx0 + βx = 0 for the function x = x(t) with t > 0, and with α and β two real parameters. • Show that the change of variables t = eu transforms the Euler equation into a second order linear equation with constant coefficients for the function x = x(u). • Describe the behavior of the solutions of the equations obtained in this way, depending on the real parameters α and β. Solution. Observe that eu is a one-to-one map from R to R+ and thus a viable substitution for d2 x t > 0. We want to compute du 2 . First, by the chain rule, dx dx dt dx = = eu , du dt du dt We differentiate again to find d2 x = eu du2 dx d2 x + eu 2 dt dt = dx d2 x + e2u 2 . du dt 2 Now we recall that eu = t and use the ODE to replace t2 ddt2x d2 x dx dx u dx − αe + βx = (1 − α) − βx. = du2 du dt du This linear second order ODE indeed has constant coefficients. We know how to solve this using the characteristic polynomial r2 + (α − 1)r + β. Let 2 1−α ∆= −β 2 1−α √ ± ∆. r± = 2 If ∆ > 0, the two linear independent solutions to the transformed equation are er+ u , er− u . Undoing the change-of-variables, we obtain the general solution C1 tr+ + C2 tr− to the original Euler equation in this case. We see that, as t → ∞, a generic solution will go to infinity/a constant/zero if r+ is positive/zero/negative. Similarly, if ∆ = 0, so r+ = r− , we obtain the general solution C1 tr+ + C2 tr+ log t and as t → ∞, a generic solution will go to infinity/zero if r+ is non-negative/negative. When ∆ < 0, we obtain C1 t(1−α)/2 cos(∆ log t) + C2 t(1−α)/2 sin(∆ log t) whose growth at infinity is determined by the sign of 1 − α. Note also that these last solutions are oscillatory. Problem 2 Problem 2.1. Solve the following differential equation: y 00 − 2y 0 + y = et /(1 + t2 ) Solution. The characteristic equation of the corresponding homogenous equation is r2 − 2r + 1 = (r−1)2 = 0, which has the single repeated root r = 1, and so y1 (t) = et , y2 (t) = tet are fundamental solutions. Observe that W (y1 , y2 )(t) = et (et + tet ) − tet et = e2t . The method of variation of parameters shows that a particular solution of the stated (nonhomogeneous) differential equation is 1 2 Z y2 (t) · et /(1 + t2 ) y1 (t) · et /(1 + t2 ) dt + y2 (t) dt W (y1 , y2 )(t) W (y1 , y2 )(t) Z dt t t dt + te = 2 1+t 1 + t2 t 2 e log(t + 1) = tet arctan t − 2 (with constant terms omitted). We conclude that the general solution of the given equation is Z Y (t) = −y1 (t) Z −et y(t) = c1 et + c2 tet + tet arctan t − et log(t2 + 1) , 2 where c1 , c2 are constants. Problem 2.2. Solve the following differential equation: y 00 − y 0 − 2y = 2e−t Solution. The characteristic equation of the corresponding homogeneous equation is r2 − r − 2 = (r − 2)(r + 1), which has roots r = −1, 2, and so the general solution of this equation is y1 (t) = c1 e−t + c2 e2t , for constants c1 , c2 . We now use the method of ”undetermined coefficients”. Since −1 is a simple root of the characteristic equation, we are looking for a particular solution of the form Y (t) = Ate−t with A some constant A. Then Y 0 (t) = Ae−t − Ate−t , Y 00 (t) = −Ae−t − Ae−t + Ate−t = Ate−t − 2Ae−t , and so Y 00 (t) − Y 0 (t) − 2Y (t) = Ate−t − 2Ae−t − Ae−t + Ate−t − 2Ate−t = −3Ae−t . It follows that if Y 00 (t) − Y 0 (t) − 2Y (t) = 2e−t , then −3A = 2, whence A = −2/3. Indeed, it is easily verified that Y (t) = − 23 te−t is a solution to the differential equation, and so the general solution is 2 y(t) = c1 e−t + c2 e2t − te−t , 3 for constants c1 , c2 . Problem 3 Let y1 (t) and y2 (t) be two solutions of the homogeneous second order equation y 00 + p(t)y 0 + q(t)y = 0 where p(t) and q(t) are continuous on an interval t ∈ I = (α, β). • If the Wronskian of the two solutions is constant, what can one say about p(t) and q(t)? • Show that if y1 (t) and y2 (t) vanish at the same point in the interval I, or if they have a maximum or a minimum at the same point, then they are not a fundamental set of solutions. Solution. Assume the Wronskian of the two solutions y1 (t), y2 (t) is a constant c. If c = 0, then y1 (t), y2 (t) are linearly dependent, and there is nothing we can say about p(t), q(t). If c 6= 0, then by Abel’s theorem W 0 + p(t)W = 0, thus we have p(t)c = 0, that is p(t) = 0. Next, suppose that y1 (t) and y2 (t) vanish at the same point t1 , then W (y1 , y2 )(t1 ) = y1 (t1 )y20 (t1 ) − y10 (t1 )y2 (t1 ) = 0. 3 Suppose y1 (t) and y2 (t) have a maximum or a minimum at the same point t1 , then y10 (t1 ) = y20 (t1 ) = 0, and then W (y1 , y2 )(t1 ) = 0. In either case the Wronskian is 0 at t1 , therefore y1 , y2 do not form a fundamental set of solutions. Problem 4 For the following differential equations describe the equilibrium solutions and the asymptotic behavior of the other solutions, for different choices of the initial condition y(0) = y0 : • dy/dt = ey − 1, with initial conditions −∞ < y0 < ∞; • dy/dt = y(a − y 2 ), for different possible values of the parameter a > 0, a = 0, or a < 0, and with initial conditions −∞ < yo < ∞. Problem 4.1. Solution. First, we find the equilibrium solutions. Solve dy = ey − 1 = 0 dt which has solution y = 0, thus this is the only equilibrium solution. Since dy/dt > 0 and is increasing for y > 0, if y0 > 0, then y → ∞ as t → ∞. Similarly, dy/dt < 0 and is decreasing for y < 0, so if y0 < 0, then y → −∞ as t → ∞. So y = 0 is an unstable equilibrium solution. Problem 4.2. Solution. If a ≤ 0, dy = y(a − y 2 ) = 0 ⇒ y = 0. dt So y = 0 is the unique equilibrium solution. Note dy/dt > 0 for y < 0 and dy/dt < 0 for y > 0. So y → 0 as t → ∞ for all initial conditions. The equilibrium solution y = 0 is asymptotically stable. If a > 0, √ dy = y(a − y 2 ) = 0 ⇒ y = 0, ± a. dt √ So y = 0, ± a are equilibrium solutions. We plot dy/dt against y: From the plot, we see that √ √ (1) for initial conditions y0√< − a, y → − a as √ t → ∞, (2) for initial conditions − a < y0√< 0, y → − a as t → ∞, √ (3) for initial conditions 0 < y√ < a, y → a as t → ∞, 0 √ (4) for initial conditions y0 > a, y → a as t → ∞. √ So the equilibrium solutions y = ± a are asymptotically stable, while y = 0 is unstable.
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