# 3: Second Order Linear Differential Equations

- Page ID
- 398

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

- 3.1: Homogeneous Equations with Constant Coefficients
- In this section we will be investigating homogeneous second order linear differential equations with constant coefficients, which can be written in the form: \[ ay'' + by' + cy = 0 \]

- 3.2: Complex Roots of the Characteristic Equation
- We have already addressed how to solve a second order linear homogeneous differential equation with constant coefficients where the roots of the characteristic equation are real and distinct. We will now explain how to handle these differential equations when the roots are complex.

- 3.4: Method of Undetermined Coefficients
- Up to now, we have considered homogeneous second order differential equations. In this discussion, we will investigate second order linear differential equations.

- 3.3: Repeated Roots and Reduction of Order
- Now that we know how to solve second order linear homogeneous differential equations with constant coefficients such that the characteristic equation has distinct roots (either real or complex), the next task will be to deal with those which have repeated roots.

- 3.5: Variation of Parameters
- In the last section we solved nonhomogeneous differential equations using the method of undetermined coefficients. This method fails to find a solution when the functions g(t) does not generate a UC-Set and we must use another approach. The approach that we will use is similar to reduction of order. Our method will be called variation of parameters.

- 3.6: Linear Independence and the Wronskian
- This is a system of two equations with two unknowns. The determinant of the corresponding matrix is the Wronskian. Hence, if the Wronskian is nonzero at some t0 , only the trivial solution exists. Hence they are linearly independent.

- 3.7: Uniqueness and Existence for Second Order Differential Equations
- To solve a second order differential equation, it is not enough to state the initial position. We must also have the initial velocity. One way of convincing yourself, is that since we need to reverse two derivatives, two constants of integration will be introduced, hence two pieces of information must be found to determine the constants.

Larry Green (Lake Tahoe Community College)