In this unit we learn how to solve constant coefficient second order linear
differential equations, and also how to interpret these solutions when the DE is
modeling a physical system. The language and ideas we introduced for first order
linear constant coefficient DE’s carry forward to the second order case—in
particular, the breakdown into the homogeneous and inhomogeneous cases; the
input-response language for the inhomogeneous case; and the general form of
solutions as
\(x = x_h + x_p\)
where \(x_h\) is the general solution to the homogeneous equation, and xp is any
particular solution to the inhomogeneous equation.
We first learn how to solve the homogeneous equation
\(ax’’ + bx’ + cx = 0\)
As in the first order case, the solutions will be exponential functions. In the
second order case, however, the exponential functions can be either real or
complex, so that we need to use the complex arithmetic and complex exponentials
we developed in the last unit.
For the second order inhomogeneous DE
\(ax’’ + bx’ + cx = f(t)\)
we will concentrate much of our attention on the important case of a sinusoidal
signal \(f(t) = B \cos(\omega t)\), and again see how we can simplify the
calculations and achieve a better understanding of the solution methods and the
solutions themselves by using complex exponentials and arithmetic.
We will run an interpretation of the results as the behavior of a physical
system in parallel with the mathematical methods and formulas, in order to have
a concrete way to understand and visualize what these results mean. We will use
a spring-mass mechanical system for the most part, since it is easy to picture.
At the end of the unit we will also explore the application of this theory to
electrical circuits, which is another very important use of these mathematical
methods.
Last Modification: 2020-09-16 Wed 21:03
- Captured On
- [2019-10-11 Fri 16:58]
- Source
- Unit II: Second Order Constant Coefficient Linear Equations |
Differential Equations | Mathematics | MIT OpenCourseWare
Captured On [2019-10-08 Tue 11:29] Source Modes and the Characteristic Equation | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 In this system, when is the equilibrium point and what does it mean? 1.1 Front In this system, when is the equilibrium point and what does it mean?
1.2 Back It’s at \(x=0\), and it’s the point where the spring is relaxed, which means that it is exerting no force....
Captured On [2019-10-11 Fri 17:46] Source [[https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-ii-second-order-constant-coefficient-linear-equations/damped-harmonic-oscillators/index.htm][Damped Harmonic Oscillators | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare]]
1 What is a damped sinusoid equation? 1.1 Front What is a damped sinusoid equation?
Write the general form
1.2 Back \(y(t) = A e^{- \lambda t} \cos(\omega t - \phi)\)
2 What is the basic solutions of DE is we get \(z(t)\) with complex exponential?...
Captured On [2019-10-16 Wed 13:53] Source Exponential Resonse 1 Can we use the superposition principle for second order equation? 1.1 Front Can we use the superposition principle for second order equation?
i.e. \(m\ddot{x} + b\dot{x} + kx = F_{ext}\)
1.2 Back Yes, you can. Suppose \(x_p\) is any solution to this equation, and \(x_h\) is the solution of the homogeneous second order equation (\(m\ddot{x} + b\dot{x} + kx = 0\))...
Captured On [2019-10-25 Fri 12:58] Source Gain and Phase Lag | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 When can we say that a system is stable? 1.1 Front When can we say that a system is /stable/?
1.2 Back If the systems’s long-term behaviour does not depend significantly on the initial conditions.
2 Give an example of stable system in mechanics?...
Captured On [2019-10-28 Mon 13:23] Source Undetermined Coefficients | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the undetermined coefficients theorem? 1.1 Front What is the undetermined coefficients theorem?
Let \(p(D)y = q(x)\), where \(q(x)\) is a polynomial
1.2 Back \(q(x)\) is a polynomial with the form \(q(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0\), where the largest \(k\) is the degree of polynomial which \(a_k \neq 0\)...
Captured On [2019-10-29 Tue 13:41] Source Linear Operators, Linear Time Invariance | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the name of this symbol \(p(D)\)? 1.1 Front What is the name of this symbol $p(D)$?
Where \(D\) is a differential operator
1.2 Back \(p(D)\) is a polynomial operator
2 What is a polynomial differential operator with constant coefficients?...
Captured On [2019-10-31 Thu 13:31] Source Pure Resonance | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare
Captured On [2019-11-11 Mon 12:39] Source Frequency Response and Practical Resonance | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the amplitude response of the system? 1.1 Front What is the amplitude response of the system?
1.2 Back It’s called to the gain of the system in terms of input angular frequency
2 What is the phase response of the system?...
Captured On [2019-11-14 Thu 13:11] Source Applications: LRC Circuits | Unit II: Second Order Constant Coefficient Linear Equations | Differential Equations | Mathematics | MIT OpenCourseWare