Math 251 Supplemental Exercises PDF

Math 251 Supplemental Exercises 1.1 Problems S1 - S7 explore a method in which we seek solutions of a specific form: S1. Find all (integer) values of n for which y(x) = xn is a solution to: 3y xy ′′...

Math 251 Supplemental Exercises 1.1 Problems S1 - S7 explore a method in which we seek solutions of a specific form: S1. Find all (integer) values of n for which y(x) = xn is a solution to: 3y xy ′′ − y ′ − =0 x S2. Find all (integer) values of n for which f (t) = tn is a solution to: d2 f 1 df 2 + =0 dt t dt S3. Find all equilibrium or constant solutions to: (i.e. find all values of C for which y(t) = C is a solution - where C is a constant) y ′′ = ecos t y ′ − p 2ty ′ + y 2 − t2 S4. Find all equilibrium or constant solutions to: (i.e. find all values of C for which φ(q) = C is a solution - where C is a constant) dφ dφ = q3φ − dq dq S5. Find all values of ω for which y = cos(ωt) is a solution to: y ′′ + 25y = 0 S6. Find all values of r for which y = ert is a solution to: y ′′ + 4y ′ + 3y = 0 S7. Find all equilibrium or constant solutions to: (i.e. find all values of C for which y(t) = C is a solution - where C is a constant) √ ′ ′ t 2 t yy y (e + cos(3t − 4)) − y e − = −et 3 + arctan(t2 + 1)  1.4 S1. First find all constant solutions, then find an explicit solution to the separable ODE : y ′ (t) = y 2 t2 What is the specific solution that satisfies the initial condition y(0) = 3 ? What is the specific solution that satisfies the initial condition y(0) = 0 ? S2. Solve the IVP explicitly: dx (2x + 6) = −t sin(t2 ) x(0) = −4 dt S3. First find all constant solutions, then solve explicitly: dz 2 − z 2 θ3 eθ = 7z 2 dθ 1.5 S1. Which of the ODE’s listed is linear? df dx y ′ (t) − 2e−t y = 4t − 1 p I) + ez f = z 2 II) III) − 2t2 x = cos(πt) dz dt dφ IV) 2ty ′ (t) = 1 V) φ + 9φ = 3t2 VI) x′ (t) − 2t sin(x) = et dt 2.2 S1. Consider a population of chipmunks which satisfies the logistic equation: P ′ (t) = P (M − P ) in which M is a constant representing the amount of food available to the chipmunks. a) If you begin with P (0) = 10 chipmunks, what is the minimum amount of food you must provide to ensure that the population will grow? (Hint: use a phase diagram.) b) Solve the ODE using the initial condition P (0) = 10 and the M value you found in part a). Is your solution increasing for all t as predicted in part a)? Page 2  2.3 S1. Suppose a boat moves through the water at speed 100 meters/sec. The engine suddenly stops, and while the boat is coasting, the resistive force of the water is proportional to the square root of the velocity, and after 2 seconds the boat is going 64 meters/sec. Set up the ODE for the velocity v(t) and clearly state the initial condition. Then solve for v(t) and then find how far the boat will coast in total. 3.3 S1. Assume b2 − 4ac < 0 in ax2 + bx + c, what is the imaginary part of the root? S2. Assume b2 − 4ac < 0 in ax2 + bx + c, show that the 2 roots must be complex conjugates. S3. Solve the IVP and graph the solution: y ′′ + 2y ′ + 2y = 0 y(0) = 0, y ′ (0) = 1 While the solution is not a periodic function, it does cross the axis at regular intervals. So find all the roots of the solution, and compute the distance (time) between successive roots to see that it is a constant. 3.4 S1. Suppose a mass-spring-dashpot system is critically damped. Let r1 and r2 be the roots of the characteristic equation of the ODE for the position function x(t). Assume that the mass begins moving at time t = 0. a) Write down the general solution to the ODE in terms of r1 and r2 , using c1 and c2 for the unknown constants in the general solution. b) Show that the mass can pass through the equilibrium position at most one time. S2. Suppose a mass-spring-dashpot system is overdamped. Let r1 and r2 be the roots of the characteristic equation of the ODE for the position function x(t). Assume that the mass begins moving at time t = 0. a) Write down the general solution to the ODE in terms of r1 and r2 , using c1 and c2 for the unknown constants in the general solution. b) Show that the mass can pass through the equilibrium position at most one time. Page 3  S3. Suppose a mass-spring-dashpot system is underdamped. Let r1 = α + iβ and r2 = α − iβ be the roots of the characteristic equation of the ODE for the position function x(t). Assume that the mass begins moving at time t = 0. a) Write down the general solution to the ODE in terms of α and β, using c1 and c2 for the unknown constants in the general solution. b) Show that the mass must pass through the equilibrium position infinitely many times. S4. Suppose a mass - spring system has position function y(t) = 6e−2t − 3te−2t. When y(t) > 0 the spring is stretched, and the spring is compressed when y(t) < 0. Determine when the mass passes through the equilibrium position, when it is maximally com- pressed, and then describe what it does after its max compression. S5. A mass-spring-dashpot system has position function x(t) = 2e−3t − 5e−2t , in which positive values of x mean the spring is stretched and negative values mean the spring is compressed. Assume the mass only began moving at t = 0. Support your answers mathematically. a) Will this mass pass through the equilibrium position? b) When is the spring most stretched? c) When is the spring most compressed? S6. Suppose a mass-spring system is described by the ODE: y ′′ + 9y = 0, with initial conditions y(0) = −1, and y ′ (0) = 0. a) How far from equilibrium will the mass get? b) Will the mass move slower as it passes through equilibrium the 4th time it passes through compared to the 1st time? S7. Suppose that the mass in a spring-mass-dashpot system with m = 1, c = 4, and k = 5 is set in motion (at t = 0) with x(0) = 2 and x′ (0) = −2. When x(t) > 0 the spring is stretched, and the spring is compressed when x(t) < 0. a) Find the position of the mass for all t > 0. b) List all the times the mass changes direction (hits a local max or min). Which of these times represents when is the mass most compressed? Page 4  S8. Suppose that the mass in a spring-mass-dashpot system with m = 1, c = 3, and k = 2 is set in motion (at t = 0) with x(0) = 0 and x′ (0) = −2. When x(t) > 0 the spring is stretched, and the spring is compressed when x(t) < 0. a) Find the position of the mass for all t > 0. b) Explain how you can tell that the spring is never stretched. (Hint: Where is the mass and which way is it moving initially, and does it ever cross the equilibrium position?) S9. Suppose that the mass in a spring-mass-dashpot system with m = 1, c = 4, and k = 4 is set in motion (at t = 0) with x(0) = 1 and x′ (0) = 2. When x(t) > 0 the spring is stretched, and the spring is compressed when x(t) < 0. a) Find the position of the mass for all t > 0. b) Explain how you can tell that the spring is never compressed. (Hint: Where is the mass and which way is it moving initially, and does it ever cross the equilibrium position?) c) When is the mass farthest away from the equilibrium position? S10. Suppose that the position of a mass, x(t), in a spring-mass-dashpot system is modeled by the ODE: x′′ + cx′ + 9x = 0 x(0) = 1, x′ (0) = 0 When x(t) > 0 the spring is stretched, and the spring is compressed when x(t) < 0. a) For what values of c will the mass oscillate and pass through the equilibrium position infinitely many times? b) Suppose c = 6, and that you are free to change the initial velocity x′ (0) to anything you like. Can you make the mass oscillate and pass through the equilibrium position infinitely many times by choosing a different value for x′ (0)? S11. Suppose that the mass in a spring-mass-dashpot √ system with m = 1, c = 2, and k = 8 is set in motion (at t = 0) with x(0) = 0 and x′ (0) = 7. When x(t) > 0 the spring is stretched, and the spring is compressed when x(t) < 0. a) Find the position of the mass for all t > 0. b) Even though the motion is not periodic, show that the mass passes through the equilibrium π position every √ seconds (which is half the “psuedo-period”). 7 Page 5  3.6 S1. Suppose you are attempting to demonstrate resonance to a large audience, and you have 3 masses (m1 < m2 < m3 ) that you will attach to a spring, which will be given a driving force. The position of the mass is modeled (in mks units) by the ODE : mx′′ + 25x = 17 cos(2πt). You know that one of these masses will produce resonance when used in this system, but you forgot which one. You know m2 = 1 (kg), and when you attached it to the system, it does not undergo resonance. This is not well received by the audience. Which mass should you pick next to regain their confidence and finally show the effect of resonance? 7.5 S1. Solve the IVP: y ′′ + 9y = u(t − 1) + δ(t − 1) y(0) = 0, y ′ (0) = 0 S2. Solve the IVP: y ′′ + 2y ′ + 2y = 3t u(t − 1) y(0) = 0, y ′ (0) = 0 Page 6