## 17Calculus Differential Equations - Chebyshev's Equation

##### 17Calculus

On this page, we discuss a special type of differential equation with polynomial coefficients called Chebyshev's Equation.

General Form: $$(1-t^2)y'' - ty' + ay = 0$$

Classification: second-order, linear, homogeneous

Chebyshev's Differential Equations

Chebyshev's equations are of the form $$\displaystyle{ (1-t^2)y'' - ty' + ay = 0 }$$ where a is a real constant. These seem specialized but they occur so often, they are worth discussing separately. These equations can be solved by using the substitution $$t = \cos( \theta )$$. This changes the differential equation into a form that can be solved more easily. See the practice problems for examples.

Practice

Unless otherwise instructed, solve these Chebyshev differential equations using the techniques on this page.

$$(1-x^2)u'' - xu' + v^2u = 0$$; use $$x = \cos(\theta)$$

Problem Statement

$$(1-x^2)u'' - xu' + v^2u = 0$$; use $$x = \cos(\theta)$$

$$u = A\cos(v\arccos(x)) + B\sin(v\arccos(x))$$

Problem Statement

$$(1-x^2)u'' - xu' + v^2u = 0$$; use $$x = \cos(\theta)$$

Solution

### Dr Chris Tisdell - 621 video solution

video by Dr Chris Tisdell

$$u = A\cos(v\arccos(x)) + B\sin(v\arccos(x))$$

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