17calculus 17calculus
First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Integrating Factors (Linear)
Exact Equations
Integrating Factors (Exact)
Constant Coefficients
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

17calculus > differential equations > polynomial coefficients


Topics You Need To Understand For This Page

Differential Equations Alpha List


Related Topics and Links

Differential Equations With Polynomial Coefficients





polynomial coefficients

\( r(t)y'' +p(t)y' + q(t)y = 0 \)

On this page, we discuss differential equations with polynomial coefficients of the form \(\displaystyle{ r(t^n)y^{(n)}(t) + a_{n-1}t^{n-1}y^{n-1}(t) + \cdots + a_0y(t) = g(t) }\), where \(r(t^n)\) is an nth-order polynomial.

For now, we will stick with these two special types of second-order homogeneous equations.

Cauchy-Euler Equations

\(\displaystyle{ t^2y'' +aty' + by = 0 }\)

Chebyshev's Equations

\(\displaystyle{ (1-t^2)y'' - ty' + ay = 0 }\)

Cauchy-Euler Differential Equation

A Cauchy-Euler Differential Equation (also called Euler-Cauchy Equation or just Euler Equation) is an equation with polynomial coefficients of the form \(\displaystyle{ t^2y'' +aty' + by = 0 }\).
These may seem kind of specialized, and they are, but equations of this form show up so often that special techniques for solving them have been developed. We will look at a couple of techniques on this page and direct you to other techniques on other pages.

this page


other pages



reduction of order

trial solution


integrating factors

Technique 1 - - Substitution

A good substitution for this type of equation is \(x=\ln(t)\). This substitution converts the differential equation into one with constant coefficients.

Technique 2 - - Trial Solution

A second technique involves using a trial solution \(y=t^k\) where k is a constant.

This video takes you through a general equation and then through 3 examples.

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. Again, 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.

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Practice Problems

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

Level B - Intermediate

Practice B01




Level C - Advanced

Practice C01

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



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