text: {[ promptMessage ]}
  
  Bookmark it
  
  {[ promptMessage ]}
  
  08SpringHwk01
  
  # 08SpringHwk01 - Math 3301 Homework Set 1 10 Points Basics...
  
  This preview shows page 1. Sign up to view the full content.
  
  This is the end of the preview. Sign up to access the rest of the document.
  
  Unformatted text preview: Math 3301 Homework Set 1 10 Points Basics Sketch the direction field for each of the following differential equations. Based on your direction field sketch determine the behavior of the solution, y ( t ) , as t (i.e. the long term behavior). If this behavior depends upon the value of y ( 0 ) give this dependence. 1. dy = 6 y 2 + y 3 - y 4 dt 2. y = ( y - 5 ) ln y , y > 0 Linear Differential Equations For problems 3 & 4 solve the given IVP. 3. x 2 + 4 y = cos ( 3 x ) - 2 xy 4. t 3 y + 4t 2 - 3t 3 y = e5t - 4e3t ( ) y ( 0 ) = 3 y ( 1 ) = -1 2 t > 0 ( ) 5. It is known that the solution to the following differential equation will have a relative minimum at t = 2 . Assuming that the solution and its derivative exist and are continuous for all t determine the value of the solution at this point. Note that because you don't have an initial condition you can't actually solve this differential equation. It is still possible however to answer this question. y - 3 y = 5 + 2e 4 t Hint : Recall from Calc I where relative extrema may occur and don't forget the differential equation, Just because you can't solve the differential equation doesn't mean that it's not needed! 6. Find the solution to the following IVP in terms of y0 . Determine the value of y0 for which the solution will have a relative maximum of y = -5 . y + 3 y = 4 - 3t 2 y ( 0 ) = y0 Hint : Once you have the solution in terms of y0 you already know what the relative maximum is, if there was just a way (cough, cough, #5....) to determine when it occured the rest should be pretty easy. 7. Find the solution to the following IVP in terms of . Find all possible long term behaviors of the solution at t . If this behavior depends on the value of give this dependence. y - 4 y ( t ) = 6 + 10e3t y ( 0) = 4 - 2 ...
  View Full Document
  
  {[ snackBarMessage ]}
  
  Ask a homework question - tutors are online
explanation: The document appears to be a homework assignment for a math course, specifically Math 3301. It contains several problems related to differential equations, including tasks to sketch direction fields, solve initial value problems (IVPs), and analyze the behavior of solutions. The document contains multiple questions but no answers.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: {[ promptMessage ]}
  
  Bookmark it
  
  {[ promptMessage ]}
  
  08SpringHwk01
  
  # 08SpringHwk01 - Math 3301 Homework Set 1 10 Points Basics...
  
  This preview shows page 1. Sign up to view the full content.
  
  This is the end of the preview. Sign up to access the rest of the document.
  
  Unformatted text preview: Math 3301 Homework Set 1 10 Points Basics Sketch the direction field for each of the following differential equations. Based on your direction field sketch determine the behavior of the solution, y ( t ) , as t (i.e. the long term behavior). If this behavior depends upon the value of y ( 0 ) give this dependence. 1. dy = 6 y 2 + y 3 - y 4 dt 2. y = ( y - 5 ) ln y , y > 0 Linear Differential Equations For problems 3 & 4 solve the given IVP. 3. x 2 + 4 y = cos ( 3 x ) - 2 xy 4. t 3 y + 4t 2 - 3t 3 y = e5t - 4e3t ( ) y ( 0 ) = 3 y ( 1 ) = -1 2 t > 0 ( ) 5. It is known that the solution to the following differential equation will have a relative minimum at t = 2 . Assuming that the solution and its derivative exist and are continuous for all t determine the value of the solution at this point. Note that because you don't have an initial condition you can't actually solve this differential equation. It is still possible however to answer this question. y - 3 y = 5 + 2e 4 t Hint : Recall from Calc I where relative extrema may occur and don't forget the differential equation, Just because you can't solve the differential equation doesn't mean that it's not needed! 6. Find the solution to the following IVP in terms of y0 . Determine the value of y0 for which the solution will have a relative maximum of y = -5 . y + 3 y = 4 - 3t 2 y ( 0 ) = y0 Hint : Once you have the solution in terms of y0 you already know what the relative maximum is, if there was just a way (cough, cough, #5....) to determine when it occured the rest should be pretty easy. 7. Find the solution to the following IVP in terms of . Find all possible long term behaviors of the solution at t . If this behavior depends on the value of give this dependence. y - 4 y ( t ) = 6 + 10e3t y ( 0) = 4 - 2 ...
  View Full Document
  
  {[ snackBarMessage ]}
  
  Ask a homework question - tutors are online
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document appears to be a homework assignment for a math course, specifically Math 3301. It contains several problems related to differential equations, including tasks to sketch direction fields, solve initial value problems (IVPs), and analyze the behavior of solutions. The document contains multiple questions but no answers.
  
  Label: multiple-question