Differential equations MOC

Cauchy-Euler differential equations

A Cauchy-Euler ODE or equidimensional ODE is a linear ODE in which each term has equal dimension of the independent variable.

Second order

In the second-order, they have the standard form

𝑥2𝑦″+𝑎𝑥𝑦′+𝑏𝑦=0

Solutions are found by finding the roots of the quadratic

𝑚2+(𝑎−1)𝑚+𝑏=0

and then

1. If 𝑚1 ≠𝑚2 in general,

𝑦=𝐶1𝑥𝑚1+𝐶2𝑥𝑚2

2. In the case 𝑚1 =𝑚∗2 and 𝑚1,𝑚2 ∉ℝ this may be rewritten as

𝑦=𝐶1𝑥𝑎cos∘ln⁡(𝑥𝑏)+𝐶2𝑥𝑎sin∘ln⁡(𝑥𝑏)

3. If 𝑚 is a repeated root

𝑦=𝐶1𝑥𝑚+𝐶2𝑥𝑚ln⁡𝑥

Practice problems

• Libretext: §2.5 Cauchy-Euler equations (worked examples)
• 2017. Elementary differential equations and boundary value problems, p. 133 (§3.4 problems 31–34)

---

tidy | en | SemBr | review