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Week 13, Long Day: Second-Order Differential Equations

So far we have only considered first-order differential equations. This week we will step it up to second-order differential equations where both first and second derivatives of the unknown function appear. We will keep the coefficients to the unknown function and its derivatives as real constants.

We will be considering both homogeneous and inhomogeneous differential equations of this type. Again today the structural theorem is really useful - in fact it turns out to be crucial. And we will see how such higher-order differential equations can be rewritten to a first-order system of differential equations so that our methods from last week can be used for essentially any order of differential equation, which will neatly tie the knot on this topic in the course.

Today’s Key Concepts

Second-order linear differential equations with constant coefficients. Solution structure. The guessing methods. The complex guessing method. Initial conditions. Existence and uniqueness of solutions. Modelling with second-order differential equations.

Preparation and Syllabus

Today’s topic covers eNote 18 Linear Second-Order Differential Equations.

Maple Ressources

  • dsolve is again useful for not just 1-st order but any order of differential equations or systems of them.
  • D allows for adding derivatives as initial conditions within dsolve.

Today’s Maple demo is SecondOrdDiffEquat and we will in exercise 4 once again use an interactive eMaple sheet for a real-life modelling problem.

Activity Program

  • 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: Your teachers are present in the study areas

Group Exercises

  1. Homogeneous Second-Order Differential Equations
  2. Second-Order Differential Equations with Initial Conditions
  3. The Structural Theorem
  4. Modelling of a Physical Situation
  5. Uniqueness of the Solution. Theory
  6. Structure of Solutions. Theory
  7. Complex Guessing Method. Theory
  8. From the Solution to the Differential Equation