You CAN Ace Differential Equations

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations 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, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

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.

General Form: $$r(t)y'' + p(t)y' + q(t)y = 0$$

Classification: second-order, linear, homogeneous

$$r(t)$$, $$p(t)$$ and $$q(t)$$ are polynomials

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 $$y=\ln(t)$$. 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. The equations developed in this video are used in several practice problems below.

### Houston Math Prep - Cauchy-Euler Differential Equations (2nd Order) [12mins-56secs]

Okay, before we go on to another type of differential equation with polynomial coefficients, let's work some practice problems.

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

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

Problem Statement

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

$$\displaystyle{ u = x^{-1} \left[ A\cos((1/2)\ln(x)) + B\sin((1/2)\ln(x)) \right] }$$

Problem Statement

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

Solution

### 620 solution video

video by Dr Chris Tisdell

$$\displaystyle{ u = x^{-1} \left[ A\cos((1/2)\ln(x)) + B\sin((1/2)\ln(x)) \right] }$$

$$2t^2y'' + 3ty' + 5y = 0$$

Problem Statement

$$2t^2y'' + 3ty' + 5y = 0$$

Solution

### 2228 solution video

$$t^2y'' + 5ty' + 4y = 0$$

Problem Statement

$$t^2y'' + 5ty' + 4y = 0$$

Solution

### 2229 solution video

$$x^2y'' + 7xy' + 8y = 0$$

Problem Statement

$$x^2y'' + 7xy' + 8y = 0$$

Solution

### 2230 solution video

$$9x^2y'' + 3xy' + y = 0$$

Problem Statement

$$9x^2y'' + 3xy' + y = 0$$

Solution

### 2231 solution video

$$x^2y'' - 9xy' + 28y = 0$$

Problem Statement

$$x^2y'' - 9xy' + 28y = 0$$

Solution

### 2232 solution video

$$x^2 y'' - 11xy'+85y = 0, y(1) = -3, y'(1) = -4$$

Problem Statement

$$x^2 y'' - 11xy'+85y = 0, y(1) = -3, y'(1) = -4$$

Solution

### 2291 solution video

video by MIP4U

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.

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 $$x = \cos(\theta)$$

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

Problem Statement

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

Solution

### 621 solution video

video by Dr Chris Tisdell

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