5: Linear Second Order Equations

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In this Chapter, we study a particularly important class of second order equations. Because of their many applications in science and engineering, second order differential equation have historically been the most thoroughly studied class of differential equations. Research on the theory of second order differential equations continues to the present day. This chapter is devoted to second order equations that can be written in the form $$ P_0(x)y''+P_1(x)y'+P_2(x)y=F(x). \nonumber $$ Such equations are said to be linear. As in the case of first order linear equations, this differential equation is said to be homogeneous if $F\equiv0$, or nonhomogeneous if $F \not \equiv 0$.

5.1: Homogeneous Linear Equations
This section is devoted to the theory of homogeneous linear equations.
- 5.1.1: Homogeneous Linear Equations (Exercises)
5.2: Constant Coefficient Homogeneous Equations
This section deals with homogeneous equations of the special form $$ ay''+by'+cy=0, $$ where $a$, $b$, and $c$ are constant ($a\ne0$). When you've completed this section you'll know everything there is to know about solving such equations.
- 5.2.1: Constant Coefficient Homogeneous Equations (Exercises)
5.3: Nonhomgeneous Linear Equations
This section presents the theory of nonhomogeneous linear equations.
- 5.3.1: Nonhomgeneous Linear Equations (Exercises)
5.4: 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.
- 5.4.1: The Method of Undetermined Coefficients I (Exercises)
5.5: 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.
- 5.5.1: The Method of Undetermined Coefficients II (Exercises)
5.6: 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.
- 5.6.1: Reduction of Order (Exercises)
5.7: 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.
- 5.7.1: Variation of Parameters (Exercises)

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