Worksheet: Systems of Ordinary Linear Differential Equations

In this worksheet, we will practice solving systems of linear ordinary differential equations.

Q1:

Find the general solution for the following system of ordinary differential equations: 𝑦′=𝑦+𝑦,𝑦′=4𝑦+𝑦.

  • A𝑦=π‘βˆ’π‘οŠ§οŠ§οŠ±οŠ©οοŠ¨οŠ±ο, 𝑦=2𝑐+2π‘οŠ¨οŠ§οŠ±οŠ©οοŠ¨οŠ±ο
  • B𝑦=π‘βˆ’π‘οŠ§οŠ§οŠ©οοŠ¨ο, 𝑦=2𝑐+2π‘οŠ¨οŠ§οŠ©οοŠ¨ο
  • C𝑦=π‘βˆ’π‘οŠ§οŠ§οŠ©οοŠ¨ο, 𝑦=2𝑐+2π‘οŠ¨οŠ§οŠ©οοŠ¨ο
  • D𝑦=π‘βˆ’π‘οŠ§οŠ§οŠ©οοŠ¨οŠ±ο, 𝑦=2𝑐+2π‘οŠ¨οŠ§οŠ©οοŠ¨οŠ±ο

Q2:

Find the general solution for the following system of ordinary differential equations: 𝑦=βˆ’π‘¦βˆ’π‘¦,𝑦=2π‘¦βˆ’4𝑦.

  • A𝑦=𝑐𝑒+𝑐𝑒,𝑦=𝑐𝑒+2π‘π‘’οŠ§οŠ§οŠ±οŠ¨οοŠ¨οŠ©οοŠ¨οŠ§οŠ¨οοŠ¨οŠ±οŠ©ο
  • B𝑦=𝑐𝑒+𝑐𝑒,𝑦=𝑐𝑒+2π‘π‘’οŠ§οŠ§οŠ±οŠ¨οοŠ¨οŠ±οŠ©οοŠ¨οŠ§οŠ±οŠ¨οοŠ¨οŠ±οŠ©ο
  • C𝑦=𝑐𝑒+𝑐𝑒,𝑦=𝑐𝑒+2π‘π‘’οŠ§οŠ§οŠ±οŠ¨οοŠ¨οŠ±οŠ©οοŠ¨οŠ§οŠ¨οοŠ¨οŠ©ο
  • D𝑦=𝑐𝑒+𝑐𝑒,𝑦=𝑐𝑒+2π‘π‘’οŠ§οŠ§οŠ±οŠ¨οŠ±οŠ¨οŠ§οŠ±οŠ¨οŠ±ο‘‰οŽ‘ο‘‰οŽ’ο‘‰οŽ‘ο‘‰οŽ’

Q3:

Suppose that you were tasked with creating a system of ordinary differential equations to model predator-prey dynamics. Let π‘₯ and 𝑦 denote the number of prey (e.g., rabbits) and predators (e.g., foxes), respectively, as a function of time 𝑑, where the positive numbers 𝛼, 𝛽, 𝛾, and 𝛿 represent parameters that describe how some predator and prey interact with each other. Which of the following systems of first-order nonlinear ordinary differential equations describe such a system?

  • Aπ‘₯β€²=𝑦(π›Όβˆ’π›½π‘₯),𝑦′=βˆ’π‘₯(π›Ύβˆ’π›Ώπ‘¦)
  • Bπ‘₯β€²=βˆ’π‘₯(π›Όβˆ’π›½π‘¦),𝑦′=𝑦(π›Ύβˆ’π›Ώπ‘₯)
  • Cπ‘₯β€²=π‘₯(π›Όβˆ’π›½π‘₯),𝑦′=βˆ’π‘¦(π›Ύβˆ’π›Ώπ‘¦)
  • Dπ‘₯β€²=π‘₯(π›Όβˆ’π›½π‘¦),𝑦′=βˆ’π‘¦(π›Ύβˆ’π›Ώπ‘₯)

Q4:

It is possible to convert an 𝑛-th order differential equation into an 𝑛-dimensional system of first-order differential equations. For the following 4th-order differential equation, identify the corresponding 4-dimensional system of first-order ordinary differential equations: 𝑦′′′′+π‘‘π‘¦β€²β€²β€²βˆ’2𝑦′′=βˆ’3π‘¦β€²βˆ’π‘¦=0.

Use the four new variables π‘₯=π‘¦οŠ§, π‘₯=π‘¦β€²οŠ¨, π‘₯=π‘¦β€²β€²οŠ©, and π‘₯=𝑦′′′οŠͺ to make this determination.

  • Aπ‘₯=π‘₯, π‘₯=π‘₯, π‘₯=π‘₯οŠͺ, π‘₯=π‘₯+3π‘₯+2π‘₯βˆ’π‘₯οŠͺοŠͺ
  • Bπ‘₯=π‘₯, π‘₯=π‘₯, π‘₯=π‘₯οŠͺ, π‘₯=βˆ’π‘‘π‘₯+2π‘₯+3π‘₯+π‘₯οŠͺοŠͺ
  • Cπ‘₯=π‘₯, π‘₯=π‘₯, π‘₯=π‘₯οŠͺ, π‘₯=π‘₯+3π‘₯+2π‘₯βˆ’π‘‘π‘₯οŠͺοŠͺ
  • Dπ‘₯=π‘₯, π‘₯=π‘₯, π‘₯=π‘₯οŠͺ, π‘₯=𝑑π‘₯+2π‘₯+3π‘₯+π‘₯οŠͺοŠͺ

Q5:

Solve the following system of linear differential equations using matrix methods, giving the matrix 𝑦𝑦.οŠ§οŠ¨π‘¦=𝑦+𝑦𝑦=4𝑦+π‘¦οŽ˜οŠ§οŠ§οŠ¨οŽ˜οŠ¨οŠ§οŠ¨

  • A𝑐12𝑒+π‘ο”βˆ’12ο π‘’οŠ§οŠ©οοŠ¨οŠ±οŠ©ο
  • B𝑐12𝑒+π‘ο”βˆ’12ο π‘’οŠ§οοŠ¨οŠ±ο
  • C𝑐12𝑒+π‘ο”βˆ’12ο π‘’οŠ§οŠ©οοŠ¨οŠ±ο
  • Dπ‘ο”βˆ’12𝑒+𝑐12ο π‘’οŠ§οŠ©οοŠ¨οŠ±ο

Q6:

Find the general solution for the following system of ordinary differential equations: 𝑦=3𝑦+2𝑦,𝑦=4𝑦+𝑦.

  • A𝑦=𝑐𝑒+π‘π‘’οŠ§οŠ§οŠ±οοŠ¨οŠ«ο, 𝑦=βˆ’2𝑐𝑒+π‘π‘’οŠ¨οŠ§οŠ±οοŠ¨οŠ±οŠ«ο
  • B𝑦=𝑐𝑒+π‘π‘’οŠ§οŠ§οŠ±οοŠ¨οŠ«ο, 𝑦=βˆ’2𝑐𝑒+π‘π‘’οŠ¨οŠ§οŠ±οοŠ¨οŠ«ο
  • C𝑦=𝑐𝑒+π‘π‘’οŠ§οŠ§οοŠ¨οŠ«ο, 𝑦=βˆ’2𝑐𝑒+π‘π‘’οŠ¨οŠ§οŠ±οοŠ¨οŠ«ο
  • D𝑦=𝑐𝑒+π‘π‘’οŠ§οŠ§οŠ±οοŠ¨οŠ«ο, 𝑦=βˆ’2𝑐𝑒+π‘π‘’οŠ¨οŠ§οοŠ¨οŠ«ο

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