First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Linear
Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
Constant Coefficients
Substitution
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
Wronskian
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Resonance
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

### Differential Equations Alpha List

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

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Differential Equations With Polynomial Coefficients

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second-order

linear

homogeneous

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.

 $$\displaystyle{ t^2y'' +aty' + by = 0 }$$ 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 substitution 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

$$\displaystyle{x^2u''+3xu'+\frac{5u}{4}=0}$$

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