Continuation: General Theory for Inhomogeneous ODEs. Stability Criteria for the Constant-coefficient ODEs

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Lesson Summary:

In this lesson, the focus is on inhomogeneous differential equations and their solutions. The inhomogeneous equation is of the form y'' + p(t)y' + q(t)y = f(t), where f(t) is the input signal or the driving term. The associated homogeneous equation is y'' + p(t)y' + q(t)y = 0, and its solutions are important in finding the solution to the inhomogeneous equation. The main theorem states that the solution to the inhomogeneous equation has the form Yp + Yc, where Yp is the particular solution and Yc is the complementary solution or the solution to the associated homogeneous equation. Two classical examples of inhomogeneous differential equations are the spring-mass system and the electric circuit with an inductance, resistance, and capacitance.

Lesson Description:

Continuation: General Theory for Inhomogeneous ODEs. Stability Criteria for the Constant-coefficient ODEs -- Lecture 12. More examples of second-order ODEs.

Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 26, 2008). License: Creative Commons BY-NC-SA.
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Additional Resources:
Questions answered by this video:
  • What is an inhomogeneous second-order differential equation?
  • What is the reduced equation?
  • What is a passive system?
  • What is a forced system?
  • Staff Review

    • Currently 4.0/5 Stars.
    This lesson has many more examples of real-world functions and both inhomogeneous and homogeneous second-order ODEs and their solution equations and theorems. The spring equation is used for a second-order ODE example. Solutions and stability conditions for different ODEs are also discussed. A good video for more second-order ODE work.