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Second-Order Linear Differential Equations - …
Second-Order Linear Differential EquationsA Second-Order Linear Differential equationhas the formwhere ,, , and are continuous functions. We saw in Section that Equations ofthis type arise in the study of the motion of a spring. In Additional Topics: Applications ofSecond- order Differential Equationswe will further pursue this application as well as theapplication to electric this section we study the case where , for all , in Equation 1. Such equa-tions are called homogeneouslinear Equations . Thus, the form of a Second-Order linearhomogeneous Differential equation isIf for some , Equation 1 is nonhomogeneousand is discussed in AdditionalTopics: Nonhomogeneous Linear basic facts enable us to solve homogeneous Linear Equations . The first of these saysthat if we know two solutions and of such an equation, then the Linear combinationis also a and are both solutions of the Linear homogeneous equa-tion (2) and and are any constants, then the functionis also a solution of Equation and are solutions of Equation 2, we haveandTherefore, using the basic rules for differentiation, we haveThus,is a solution of Equation other fact we need is given by the following theorem, which is proved in moreadvanced courses.
Second-Order Linear Differential Equations A second-order linear differential equationhas the form where , , , and are continuous functions. We saw in Section 7.1 that equations of this type arise in the study of the motion of a spring.
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