W2.5/13강 일원 2차 상미분 방정식(Homogeneous 2nd-order ODEs with constant coefficients)

[커세라] 이과생을 위한 미분 방정식(Differential Equations for Engineers)

0주: 환영(Welcome)
    W0.1 강좌 예고(Course Trailer)
    W0.2~5 강좌 개요(Course Overview)
    W0.6~7/1강 미분 방정식 소개 및 연습문제
                     (Introduction to Differential Equations & Practice Quiz)

1주: 1차 미분 방정식(Frst-order Differential Equation)
    W1.1/2강 수치해석:오일러 방법(Euler Method)
    W1.2/3강 1차 미분 방정식 세우기(Separable First-order equations)
    W1.3/4강 1차 미분 방정식 세우기 예제(Separable First-order equations:Example)
    W1.4 연습문제(Practice Quiz:Separable First-order ODEs)
    W1.5/5강 선형 1차 미분 방정식(Linear First-order Equations)
    W1.6/6강 선형 1차 미분 방정식:예제(Linear First-order Equations:Example)
    W1.7 연습문제(Practice Quiz:Linear First-Order ODE)
    W1.8/7강 선형 1차 미방 응용:예금(Application: Compound interest)
    W1.9/8강 선형 1차 미방 응용:종말속도(Application: Terminal Velocity)
    W1.10/9강 선형 1차 미방 응용:RC회로(Application: RC Circuit)
    W1.11 연습문제(Practice Quiz: Applications)
    W1.12 평가문제(Week-1 Graded Quiz)

2주: 2차 미분 방정식(Second-order Differential equations)
    W2.1/10강 고차 미분 방정식에 대한 오일러 방법(Euler method for higher-order odes)
    W2.2/11강 중첩의 원리(Principle of Superposition)
    W2.3/12강 론스키 행렬식(The Wronskian)
    W2.4/보강1 복소수(complex numbers)

W2.5/13강 일원 2차 상미분 방정식
               (Homogeneous 2nd-order ODEs with constant coefficients)/동영상/영문자막



We're starting our study of homogeneous second-order ODEs with constant coefficients. The general differential equation is,



Ansatz(가설풀이):

Our solution method basically guesses the solution with a "free parameter r". We guess an exponential function, exp(rt),



because the exponential function will cancel out of the equation



leaving us an algebraic equation for r.



This is called the characteristic equation of the differential equation.



Because it's a quadratic equation, one of three things can happen.

(i) We can have distinct real roots which case we have two exponential functions, e to the r plus t and e to the r minus t. We then apply the principle of superposition to that.



(ii) We can have complex conjugate roots, which case we have two complex exponential functions. We will learn how to deal with that to get two real functions.



(iii) We can have repeated roots in which case we only have one exponential function. We're going to have to do some magic to get the second solution.



2차 미분 방정식은 2개의 해를 가져야 하는데 중근(repeated root)으로 1개의 근이 1개만 존재한다면 풀이를 수월케 하려던 가설풀이 방법(Ansatz)에 오류가 있음을 뜻한다. 따라서 중근인 경우 두번째 근을 찾아내야 한다.

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