Worksheet: Cauchy–Euler Differential Equation

In this worksheet, we will practice solving Cauchy–Euler differential equations of the general form aₙ xⁿ y⁽ⁿ⁾ + aₙ₋₁ xⁿ⁻¹ y⁽ⁿ⁻¹⁾ + ... + a₀ y = f(x).

Q1:

The functions 𝑦 = π‘₯  , 𝑦 = π‘₯   , and 𝑦 = π‘₯   are three linearly independent solutions of the differential equation π‘₯ 𝑦 βˆ’ 3 π‘₯ 𝑦 + 6 π‘₯ 𝑦 βˆ’ 6 𝑦 = 0  (  )     . Find a particular solution satisfying the initial conditions 𝑦 ( 1 ) = 6 , 𝑦 ( 1 ) = 1 4  , and 𝑦 ( 1 ) = 2 2   .

  • A 𝑦 = π‘₯ + π‘₯ + 3 π‘₯  
  • B 𝑦 = π‘₯ βˆ’ 2 π‘₯ + 3 π‘₯  
  • C 𝑦 = π‘₯ + 2 π‘₯ + 3 π‘₯  
  • D 𝑦 = π‘₯ βˆ’ 2 π‘₯ βˆ’ 3 π‘₯  
  • E 𝑦 = π‘₯ βˆ’ 2 π‘₯ + π‘₯  

Q2:

The functions 𝑦 = π‘₯  , 𝑦 = π‘₯    , and 𝑦 = π‘₯ π‘₯    l n are three linearly independent solutions of the differential equation π‘₯ 𝑦 + 6 π‘₯ 𝑦 + 4 π‘₯ 𝑦 βˆ’ 4 𝑦 = 0  (  )     . Find a particular solution satisfying the initial conditions 𝑦 ( 1 ) = 1 , 𝑦 ( 1 ) = 5  , and 𝑦 ( 1 ) = βˆ’ 1 1   .

  • A 𝑦 = 2 π‘₯ βˆ’ π‘₯ + π‘₯ π‘₯     l n
  • B 𝑦 = 8 3 π‘₯ βˆ’ 8 3 π‘₯ + π‘₯ π‘₯     l n
  • C 𝑦 = 2 π‘₯ βˆ’ 2 π‘₯ + π‘₯ π‘₯     l n
  • D 𝑦 = 1 1 6 π‘₯ βˆ’ 5 6 π‘₯ + 3 2 π‘₯ π‘₯     l n
  • E 𝑦 = π‘₯ βˆ’ 2 π‘₯ + π‘₯ π‘₯     l n

Q3:

Find the general solution for the following ordinary differential equation with variable coefficients: π‘₯ 𝑦 β€² β€² + π‘₯ 𝑦 β€² + 𝑦 = π‘₯  . This is an example of an Euler-Cauchy differential equation (nonzero right-hand side).

  • A 𝑦 = 𝑐 ( ( π‘₯ ) ) + 𝑐 ( ( π‘₯ ) ) βˆ’ π‘₯ 2   c o s l n s i n l n
  • B 𝑦 = 𝑐 ( ( π‘₯ ) ) + 𝑐 ( ( π‘₯ ) ) + π‘₯ 2   c o s l n s i n l n
  • C 𝑦 = 𝑐 ( ( π‘₯ ) ) + 𝑐 ( ( π‘₯ ) ) + √ π‘₯   c o s l n s i n l n
  • D 𝑦 = 𝑐 ( ( π‘₯ ) ) + 𝑐 ( ( π‘₯ ) ) + π‘₯ 2   l n c o s l n s i n

Q4:

Find the general solution for the homogeneous ordinary differential equation with variable coefficients π‘₯ 𝑦 + 3 π‘₯ 𝑦 βˆ’ 3 𝑦 = 0     . This is an example of an Euler–Cauchy differential equation (zero right-hand side).

  • A 𝑦 = 𝑐 π‘₯ + 𝑐  
  • B 𝑦 = 𝑐 π‘₯ + 𝑐 π‘₯    
  • C 𝑦 = 𝑐 π‘₯ + 𝑐 π‘₯    
  • D 𝑦 = 𝑐 π‘₯ + 𝑐 π‘₯    

Q5:

Find the general solution for the following ordinary differential equation using the reduction of order method: π‘₯ 𝑦 β€² β€² + π‘₯ 𝑦 β€² βˆ’ 𝑦 = 0  , π‘₯ β‰  0 and 𝑦 = π‘₯  .

  • A 𝑦 = 𝑐 π‘₯ + 𝑐 π‘₯ + π‘₯ 2 ( π‘₯ )       l n
  • B 𝑦 = 𝑐 π‘₯ + 𝑐 π‘₯   
  • C 𝑦 = 𝑐 π‘₯ + 𝑐 π‘₯ + π‘₯ 2 ( π‘₯ )    l n
  • D 𝑦 = 𝑐 π‘₯ + 𝑐 π‘₯      
  • E 𝑦 = 𝑐 π‘₯ + 𝑐 π‘₯    

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