Download presentation

Presentation is loading. Please wait.

Published byGeorge Howell Modified over 2 years ago

1
Section 6.1 Cauchy-Euler Equation

2
THE CAUCHY-EULER EQUATION Any linear differential equation of the from where a n,..., a 0 are constants, is said to be a Cauchy-Euler equation, or equidimensional equation. NOTE: The powers of x match the order of the derivative.

3
We will confine our attention to solving the homogeneous second-order equation. The solution of higher-order equations follows analogously. The nonhomogeneous equation can be solved by variation of parameters once the complimentary function, y c (x), is found. HOMOGENEOUS 2 ND -ORDER CAUCHY-EULER EQUATION

4
THE SOLUTION We try a solution of the form y = x m, where m must be determined. The first and second derivatives are, respectively, y′ = mx m − 1 and y″ = m(m − 1)x m − 2. Substituting into the differential equation, we find that x m is a solution of the equation whenever m is a solution of the auxiliary equation

5
THREE CASES There are three cases to consider, depending on the roots of the auxiliary equation. Case I:Distinct Real Roots Case II:Repeated Real Roots Case III:Conjugate Complex Roots

6
Let m 1 and m 2 denote the real roots of the auxiliary equation where m 1 ≠ m 2. Then form a fundamental solution set, and the general solution is CASE I — DISTINCT REAL ROOTS

7
Let m 1 = m 2 denote the real root of the auxiliary equation. Then we obtain one solution—namely, Using the formula from Section 4.2, we obtain the solution Then the general solution is CASE II — REPEATED REAL ROOTS

8
Let m 1 = α + βi and m 2 = α − βi, then the solution is Using Euler’s formula, we can obtain the fundamental set of solutions y 1 = x α cos(β ln x) and y 2 = x α sin(β ln x). Then the general solution is CASE III — CONJUGATE COMPLEX ROOTS

9
ALTERNATE METHOD OF SOLUTION Any Cauchy-Euler equation can be reduced to an equation with constant coefficients by means of the substitution x = e t.

10
HOMEWORK 1–45 odd

Similar presentations

OK

Math 3120 Differential Equations with Boundary Value Problems

Math 3120 Differential Equations with Boundary Value Problems

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on computer virus symptoms Animated ppt on magnetism powerpoint Ppt on education for sustainable development Microsoft office ppt online viewer Ppt on aerobics shoes Ppt on amplitude shift keying remote Ppt on mcx stock exchange Ppt on current account deficit indonesia Ppt on 3-phase squirrel cage induction motor Ppt on cross site scripting owasp