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 β€² = 𝑦 + 𝑦 , 𝑦 β€² = 4 𝑦 + 𝑦 .      

  • A 𝑦 = 𝑐 βˆ’ 𝑐       , 𝑦 = 2 𝑐 + 2 𝑐      
  • B 𝑦 = 𝑐 βˆ’ 𝑐         , 𝑦 = 2 𝑐 + 2 𝑐        
  • C 𝑦 = 𝑐 βˆ’ 𝑐        , 𝑦 = 2 𝑐 + 2 𝑐       
  • D 𝑦 = 𝑐 βˆ’ 𝑐       , 𝑦 = 2 𝑐 + 2 𝑐      

Q2:

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

  • A 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 , 𝑦 = 𝑐 𝑒 + 2 𝑐 𝑒                
  • B 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 , 𝑦 = 𝑐 𝑒 + 2 𝑐 𝑒                
  • C 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 , 𝑦 = 𝑐 𝑒 + 2 𝑐 𝑒                  
  • D 𝑦 = 𝑐 𝑒 + 𝑐 𝑒 , 𝑦 = 𝑐 𝑒 + 2 𝑐 𝑒                  

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?

  • A π‘₯ β€² = π‘₯ ( 𝛼 βˆ’ 𝛽 𝑦 ) , 𝑦 β€² = βˆ’ 𝑦 ( 𝛾 βˆ’ 𝛿 π‘₯ )
  • B π‘₯ β€² = 𝑦 ( 𝛼 βˆ’ 𝛽 π‘₯ ) , 𝑦 β€² = βˆ’ π‘₯ ( 𝛾 βˆ’ 𝛿 𝑦 )
  • C π‘₯ β€² = π‘₯ ( 𝛼 βˆ’ 𝛽 π‘₯ ) , 𝑦 β€² = βˆ’ 𝑦 ( 𝛾 βˆ’ 𝛿 𝑦 )
  • D π‘₯ β€² = βˆ’ π‘₯ ( 𝛼 βˆ’ 𝛽 𝑦 ) , 𝑦 β€² = 𝑦 ( 𝛾 βˆ’ 𝛿 π‘₯ )

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 β€² β€² β€² β€² + 𝑑 𝑦 β€² β€² β€² βˆ’ 2 𝑦 β€² β€² = βˆ’ 3 𝑦 β€² βˆ’ 𝑦 = 0 .

Use the four new variables π‘₯ = 𝑦  , π‘₯ = 𝑦 β€²  , π‘₯ = 𝑦 β€² β€²  , and π‘₯ = 𝑦 β€² β€² β€² οŠͺ to make this determination.

  • A π‘₯ = π‘₯    , π‘₯ = π‘₯    , π‘₯ = π‘₯   οŠͺ , π‘₯ = π‘₯ + 3 π‘₯ + 2 π‘₯ βˆ’ π‘₯ οŠͺ    οŠͺ
  • B π‘₯ = π‘₯    , π‘₯ = π‘₯    , π‘₯ = π‘₯   οŠͺ , π‘₯ = 𝑑 π‘₯ + 2 π‘₯ + 3 π‘₯ + π‘₯  οŠͺ    οŠͺ
  • C π‘₯ = π‘₯    , π‘₯ = π‘₯    , π‘₯ = π‘₯   οŠͺ , π‘₯ = βˆ’ 𝑑 π‘₯ + 2 π‘₯ + 3 π‘₯ + π‘₯ οŠͺ    οŠͺ
  • D π‘₯ = π‘₯    , π‘₯ = π‘₯    , π‘₯ = π‘₯   οŠͺ , π‘₯ = π‘₯ + 3 π‘₯ + 2 π‘₯ βˆ’ 𝑑 π‘₯  οŠͺ    οŠͺ

Q5:

Solve the following system of linear differential equations using matrix methods, giving the matrix  𝑦 𝑦  .   𝑦 = 𝑦 + 𝑦 𝑦 = 4 𝑦 + 𝑦        

  • A 𝑐  1 2  𝑒 + 𝑐  βˆ’ 1 2  𝑒       
  • B 𝑐  βˆ’ 1 2  𝑒 + 𝑐  1 2  𝑒      
  • C 𝑐  1 2  𝑒 + 𝑐  βˆ’ 1 2  𝑒      
  • D 𝑐  1 2  𝑒 + 𝑐  βˆ’ 1 2  𝑒     

Q6:

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

  • A 𝑦 = 𝑐 𝑒 + 𝑐 𝑒       , 𝑦 = βˆ’ 2 𝑐 𝑒 + 𝑐 𝑒       
  • B 𝑦 = 𝑐 𝑒 + 𝑐 𝑒        , 𝑦 = βˆ’ 2 𝑐 𝑒 + 𝑐 𝑒        
  • C 𝑦 = 𝑐 𝑒 + 𝑐 𝑒        , 𝑦 = βˆ’ 2 𝑐 𝑒 + 𝑐 𝑒      
  • D 𝑦 = 𝑐 𝑒 + 𝑐 𝑒        , 𝑦 = βˆ’ 2 𝑐 𝑒 + 𝑐 𝑒       

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.