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3: Higher order linear ODEs

Last updated

Oct 30, 2024
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- 2.6E: Exercises for Section 2.6
- 3.1: Second order linear ODEs

Vinh Kha Nguyen & Neelam R. Shukla
Oklahoma State University

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We have already studied the basics of differential equations, including separable first-order equations. In this chapter, we go a little further and look at second-order equations, which are equations containing second derivatives of the dependent variable. The solution methods we examine are different from those discussed earlier, and the solutions tend to involve trigonometric functions as well as exponential functions. Here we concentrate primarily on second-order equations with constant coefficients.

3.1: Second order linear ODEs
This page covers second-order linear differential equations, addressing both homogeneous and nonhomogeneous types. Key concepts include the general form of these equations, the superposition principle, and the uniqueness of solutions. Techniques like reduction of order for finding second solutions are detailed, illustrated with examples such as $y_1 = x$ . The text emphasizes understanding methods over memorizing formulas and sets the stage for tackling nonhomogeneous equations.
- 3.1E: Exercises for Section 3.1
3.2: The Method of Undetermined Coefficients I
This section present the method of undetermined coefficients, which can be used to solve nonhomogeneous equations of the form ay''+by'+cy=F(x) where a, b, and c are constants and F(x) has a special form that is still sufficiently general to occur in many applications. This sections makes extensive use of the idea of variation of parameters introduced previously.
- 3.2E: Exercises for Section 3.2
3.3: The Method of Undetermined Coefficients II
In this section, we use the Method of Undetermined Coefficients to find solutions to the constant coefficient equation ay''+by'+cy=exp{λx}(P(x) cos ω x + Q(x) sin ω x) where λ and ω are real numbers, ω is not zero, and P and Q are polynomials.
- 3.3E: Exercises for Section 3.3
3.4: Constant coefficient second order linear ODEs
This page explains solving second-order linear homogeneous differential equations using the characteristic equation method, covering distinct, repeated, and complex roots. It includes the general solution forms and adaptations for double roots, along with examples. Additionally, it discusses complex numbers, their properties, and Euler's formula, applying these concepts to derive trigonometric identities.
- 3.4E: Exercises for Section 3.4
3.5: Higher order linear ODEs
The basic results about linear ODEs of higher order are essentially the same as for second order equations, with 2 replaced by nn . The important concept of linear independence is somewhat more complicated when more than two functions are involved.
- 3.5E: Exercises for Section 3.5
3.6: Reduction of Order
This section deals with reduction of order, a technique based on the idea of variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know one nontrivial (not identically zero) solution of the associated homogeneous equation.
- 3.6E: Exercises for Section 3.6
3.7: Variation of Parameters
This section deals with the method traditionally called variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know two nontrivial solutions (with nonconstant ratio) of the associated homogeneous equation.
- 3.7E: Exercises for Section 3.7
3.8: Mechanical Vibrations
Let us look at some applications of linear second order constant coefficient equations.
- 3.8E: Exercises for Section 3.8
3.9: Nonhomogeneous Equations
What about nonhomogeneous linear ODEs? For example, the equations for forced mechanical vibrations.
- 3.9E: Exercises for Section 3.9
3.10: Forced Oscillations and Resonance
Let us consider to the example of a mass on a spring. We now examine the case of forced oscillations, which we did not yet handle.
- 3.10E: Exercises for Section 3.10

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