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Exam 1 Review Problems: Differential Equations, Exams of Applied Differential Equations

Rutgers University - New BrunswickApplied Differential Equations

A set of review problems for an exam on differential equations. It covers topics such as initial-value problems, newton's law of cooling, direction fields, phase lines, and systems of differential equations. The problems are designed to help students prepare for the exam by reinforcing key concepts and problem-solving techniques.

Typology: Exams

2024/2025

Uploaded on 03/05/2025

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Download Exam 1 Review Problems: Differential Equations and more Exams Applied Differential Equations in PDF only on Docsity!

Exam 1 Review Problems

For details on exam coverage and a list of topics, please see the class website. Note that this is a set of review problems, and NOT a practice exam.

  1. Find the solution of the initial-value problem
  1. A cup of soup is initially 170◦^ F, and is left in a room with an ambient temperature of 70◦^ F. Suppose that at time t = 0 it is cooling at a rate of 20◦^ F per minute. (a) Assume Newton’s Law of Cooling applies: The rate of cooling is proportional to the difference between the current temperature and the ambient temperature. Write down an IVP that models the temperature of the soup. (b) Solve the above IVP, for the temperature T ( t ) at any time t. (c) How long does it take the soup to cool to a temperature of 110◦^ F?
  1. Problem #42, page 138 in the textbook (this is in the review exercises for Chapter 1). 5. Consider the ODE y^0 = y^3 − 3 y + a, where a is a parameter. (a) Using the graph of y^3 − 3 y , Sind the values of the parameter a where any bifurcations occur. Hint: Note that the right-hand side is just vertical translates of this graph. (b) For all qualitatively distinct cases, draw the phase line for the system. (c) Sketch a bifurcation diagram for this equation.
  1. The related problems #16 on page 50 and #11 on page 180 in the textbook.
  1. Solve the following IVP What is the largest interval, containing the initial time, on which the solution is deSined?
  1. The Sigures below show two phase planes with direction Sields. On each Sigure, sketch the trajectories (for both negative and positive times) satisfying the initial conditions (a) ( x (0) ,y (0)) = (0 , −1) (b) ( x (0) ,y (0)) = (− 3 , 1).
  2. Convert the following second-order ODE into a Sirst-order system: t^2 y^00 + ty^0 + ( t^2 − 0_._ 25) y = 0_._
  1. Find a solution of the initial-value problem (IVP) , Hint: Finding a solution does not necessarily mean Sinding an explicit solution y = y ( x ). Solutions can often be thought of curves in the (here) xy - plane. Can you Sind an explicit solution to this equation?
  1. Consider the second-order equation y^00 + sin y = 0_._ (a) Convert the above into a Sirst-order system (i.e. Y^0 = F ( Y ), where Y ∈R^2. You need to identify Y and F ( Y ).). (b) By hand, approximately draw the vector Sield obtained in part (a). Use this to then drawapproximate solution trajectories in the phase plane. Hint: The second-order equation represents the motion of a free undamped pendulum.