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17Calculus Differential Equations - Chebyshev's Equation

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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

Instructions - - 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 \( x = \cos(\theta) \)

Final Answer

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

Problem Statement

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

Solution

621 video

video by Dr Chris Tisdell

Final Answer

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

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