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‘₯π‘¦βˆ’3π‘₯𝑦+6π‘₯π‘¦βˆ’6𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(1)=6, 𝑦(1)=14, and 𝑦(1)=22.

  • A𝑦=π‘₯+2π‘₯+3π‘₯
  • B𝑦=π‘₯βˆ’2π‘₯+3π‘₯
  • C𝑦=π‘₯βˆ’2π‘₯+π‘₯
  • D𝑦=π‘₯+π‘₯+3π‘₯
  • E𝑦=π‘₯βˆ’2π‘₯βˆ’3π‘₯

Q2:

The functions 𝑦=π‘₯, 𝑦=π‘₯, and 𝑦=π‘₯π‘₯ln are three linearly independent solutions of the differential equation π‘₯𝑦+6π‘₯𝑦+4π‘₯π‘¦βˆ’4𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(1)=1, 𝑦(1)=5, and 𝑦(1)=βˆ’11.

  • A𝑦=2π‘₯βˆ’π‘₯+π‘₯π‘₯ln
  • B𝑦=π‘₯βˆ’2π‘₯+π‘₯π‘₯ln
  • C𝑦=116π‘₯βˆ’56π‘₯+32π‘₯π‘₯ln
  • D𝑦=2π‘₯βˆ’2π‘₯+π‘₯π‘₯ln
  • E𝑦=83π‘₯βˆ’83π‘₯+π‘₯π‘₯ln

Q3:

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

  • A𝑦=𝑐((π‘₯))+𝑐((π‘₯))+√π‘₯coslnsinln
  • B𝑦=𝑐((π‘₯))+𝑐((π‘₯))+π‘₯2lncoslnsin
  • C𝑦=𝑐((π‘₯))+𝑐((π‘₯))βˆ’π‘₯2coslnsinln
  • D𝑦=𝑐((π‘₯))+𝑐((π‘₯))+π‘₯2coslnsinln

Q4:

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

  • A𝑦=𝑐π‘₯+π‘οŠ§οŠ¨
  • B𝑦=𝑐π‘₯+𝑐π‘₯
  • C𝑦=𝑐π‘₯+𝑐π‘₯
  • D𝑦=𝑐π‘₯+𝑐π‘₯

Q5:

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

  • A𝑦=𝑐π‘₯+𝑐π‘₯
  • B𝑦=𝑐π‘₯+𝑐π‘₯+π‘₯2(π‘₯)ln
  • C𝑦=𝑐π‘₯+𝑐π‘₯+π‘₯2(π‘₯)ln
  • D𝑦=𝑐π‘₯+𝑐π‘₯
  • E𝑦=𝑐π‘₯+𝑐π‘₯

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