The document discusses the Cauchy-Euler equation, which is a specific type of differential equation characterized by its variable coefficients dependent on powers of x. It provides a method to transform this equation into a linear form with constant coefficients using substitutions. Several examples demonstrate how to solve different forms of the Cauchy-Euler equation.

1. DIFFERENTIAL EQUATION (MT-202) SYED AZEEM INAM
DIFFERENTIAL EQUATION (MT-202)
LECTURE #14
CAUCHY-EULER EQUATION:
The equation of the form
𝑎0 𝑥 𝑛
𝑑 𝑛
𝑦
𝑑𝑥 𝑛
+ 𝑎1 𝑥 𝑛−1
𝑑 𝑛−1
𝑦
𝑑𝑥 𝑛−1
+ ⋯ + 𝑎 𝑛−1 𝑥
𝑑𝑦
𝑑𝑥
+ 𝑎 𝑛 𝑦 = 𝐹(𝑥)
Is called Cauchy-Euler equation or equi-dimensional equation.
The equation can be reduced to a linear differential equation with constant
coefficients by the transformation
𝑥 = 𝑒 𝑡
𝑜𝑟 𝑦 = 𝑙𝑛𝑥
EXAMPLE #1: Solve: 𝑥2 𝑑2 𝑦
𝑑𝑥2
− 2𝑥
𝑑𝑦
𝑑𝑥
+ 2𝑦 = 𝑥3
EXAMPLE #2: Solve: 𝑥2 𝑑2 𝑦
𝑑𝑥2
+ 7𝑥
𝑑𝑦
𝑑𝑥
+ 5𝑦 = 𝑥5
EXAMPLE #3: Solve: 𝑥2 𝑑2 𝑦
𝑑𝑥2
− 3𝑥
𝑑𝑦
𝑑𝑥
+ 5𝑦 = sin(𝑙𝑛𝑥)
EXAMPLE #4: Solve: 𝑥3
𝑦′′′
+ 2𝑥2
𝑦′′
+ 𝑥𝑦′
− 𝑦 = 15𝑐𝑜𝑠(𝑙𝑛𝑥)