MATH251-Worksheet 2[1] - MATH 251 Work sheet / Things to...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
1 MATH 251 Work sheet / Things to know Chapter 3 1. Second order linear differential equation Standard form: What makes it homogeneous? We will, for the most part, work with equations with constant coefficients only. What is the standard form of a second order linear equation with constant coefficients? Ex . Can you think of any function(s) that satisfy each equation (w/ constant coefficients) below? (a) y ″ - 25 y = 0 (b) y ″ + 25 y = 0 (c) y ″ - 25 y ′ = 0 The example (c) above is an instance of a second order linear equation with the y ±term missing. It is essentially a first order linear equation in disguise. All equations of this type can be solved by changing it into a first order equation with the substitutions u = y and u ′ = y ″, then use the integrating factor method to solve for u , and integrate the result to find y .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 2. The characteristic equation Given the equation ay ″ + by ′ + cy = 0, what is its characteristic equation? Any root, r , of the characteristic equation has the property that y = e rt always satisfies the equation above. Therefore, y = e rt will be a particular solution for each root r . Consequently, an important formula to remember for this class is (surprisingly) the quadratic formula: a ac b b r 2 4 2 - ± - = Note that the characteristic equation method does not require the given differential equation to be put into its standard form first – the quadratic formula simply doesn’t care whether or not the leading coefficient is 1.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/16/2011 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.

Page1 / 9

MATH251-Worksheet 2[1] - MATH 251 Work sheet / Things to...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online