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ODE. What kind of fu

ODE. What kind of fu

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Only the highlighted ones are needed. All of #5. I need help 

1.) The first order linear homogeneous ODE with constant coefficients is an ODE of the form y'+ay = 0, where a is a constant. Solve this ODE. What kind of function is your solution? Be sure to remember this solution! 2.) (a) An LR circuit with an inductance of 2 H and a resistance of 50 N is driven by constant voltage of 100 V. Find the current within the circuit. Since no initial condition is given, there will be an arbitrary constant in your answer. (b) Regardless of a known initial condition, to what value does the current approach as time gets large? (c) From part a, it can be observed that the 1st order ODE is autonomous. Explain why the ODE is autonomous. Then, find the only critical point and determine its stability type. How is the critical point value related to what you got in part b? (d) Find the current using only Ohm's Law. Now suppose you have an initial current that is known (say I(0)=10). Is the description of the current you got in part d able to account for this knowledge? How about the description of the current gotten in part a? (e) Between the methods used in parts a and d, which method produced a more robust description of the current through this circuit? Explain. 3.) An RC circuit with a 10 resistor and a 0.001 F capacitor is driven by a voltage E(t) = 10 cos (60t) V. If the initial charge is 0 coulombs, determine the charge function along with the current function. Feel free to use an integral table for the integral of the exponential times the cosine function in this problem; no need to reference its usage! 4.) (a) Use Euler's method with step size h = 0.2 without using a computer such as MATLAB to approximate the solution to the initial value problem y' x = 1.2, 1.4, 1.6,, 1.8, and x = 2. уе", у(1) = 2 at the points (b) Solve the IVP to find the unknown function y. (c) Compare the approximated values found in part a to the actual values y(1.2), y(1.4), y(1.6), and y(1.8), and y(2). What do you notice about the difference between the approximated values versus the exact values as the x value increases? Why do you think this is? 5.) Suppose we wish to model the height of a projectile shot directly up at time zero at time t. Let x(t) be this function. If we assume that the only forces acting on the projectile after time zero are gravity and air resistance, one way to model the resulting motion would be the ODE m d = -mg – kv, where k is a constant of proportionality determined by the shape of the projectile and v(t) represents the velocity of the function at time t. (a) Solve the ODE for v(t) assuming that the initial velocity is the number vo. (b) Determine what happens to v(t) as t –→0. (c) This limiting value identified in part b is often called the terminal velocity. Suppose there are two objects with masses m1 and m2 with mị > m2. Using part a, which object will have the higher terminal velocity? Explain. (d) Suppose we used our old model m du = -mg which did not account for air resistance. Is there a limit to how fast an object in this model will move (i.e. under gravitational acceleration without air resistance)? Explain. What about the model in part a? Explain.
Transcribed Image Text:1.) The first order linear homogeneous ODE with constant coefficients is an ODE of the form y'+ay = 0, where a is a constant. Solve this ODE. What kind of function is your solution? Be sure to remember this solution! 2.) (a) An LR circuit with an inductance of 2 H and a resistance of 50 N is driven by constant voltage of 100 V. Find the current within the circuit. Since no initial condition is given, there will be an arbitrary constant in your answer. (b) Regardless of a known initial condition, to what value does the current approach as time gets large? (c) From part a, it can be observed that the 1st order ODE is autonomous. Explain why the ODE is autonomous. Then, find the only critical point and determine its stability type. How is the critical point value related to what you got in part b? (d) Find the current using only Ohm's Law. Now suppose you have an initial current that is known (say I(0)=10). Is the description of the current you got in part d able to account for this knowledge? How about the description of the current gotten in part a? (e) Between the methods used in parts a and d, which method produced a more robust description of the current through this circuit? Explain. 3.) An RC circuit with a 10 resistor and a 0.001 F capacitor is driven by a voltage E(t) = 10 cos (60t) V. If the initial charge is 0 coulombs, determine the charge function along with the current function. Feel free to use an integral table for the integral of the exponential times the cosine function in this problem; no need to reference its usage! 4.) (a) Use Euler's method with step size h = 0.2 without using a computer such as MATLAB to approximate the solution to the initial value problem y' x = 1.2, 1.4, 1.6,, 1.8, and x = 2. уе", у(1) = 2 at the points (b) Solve the IVP to find the unknown function y. (c) Compare the approximated values found in part a to the actual values y(1.2), y(1.4), y(1.6), and y(1.8), and y(2). What do you notice about the difference between the approximated values versus the exact values as the x value increases? Why do you think this is? 5.) Suppose we wish to model the height of a projectile shot directly up at time zero at time t. Let x(t) be this function. If we assume that the only forces acting on the projectile after time zero are gravity and air resistance, one way to model the resulting motion would be the ODE m d = -mg – kv, where k is a constant of proportionality determined by the shape of the projectile and v(t) represents the velocity of the function at time t. (a) Solve the ODE for v(t) assuming that the initial velocity is the number vo. (b) Determine what happens to v(t) as t –→0. (c) This limiting value identified in part b is often called the terminal velocity. Suppose there are two objects with masses m1 and m2 with mị > m2. Using part a, which object will have the higher terminal velocity? Explain. (d) Suppose we used our old model m du = -mg which did not account for air resistance. Is there a limit to how fast an object in this model will move (i.e. under gravitational acceleration without air resistance)? Explain. What about the model in part a? Explain.
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