Second Order Equations

So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second order linear equations. (Sort of)

Bad (or Good?) News Higher order equations are MUCH more difficult to solve than first order equations. So we start simple, and work our way up. Linear equations are hard. Nonlinear equations are crazy hard.

Linear Second Order Equations The form of a second order linear equation: Not simple enough! Still not simple enough! Homogenous equation.

Linear Second Order Equations The form of a second order linear equation: Not simple enough! Still not simple enough!

Linear Second Order Equations The form of a second order linear equation: Not simple enough! Still not simple enough! Okay, that’s a good place to start.

Second Order Linear Homogenous Equations with Constant Coefficients Side note: We already know one specific solution to this equation. Equilibrium Solution! Always check for equilibrium solutions!!!!

Second Order Linear Homogenous Equations with Constant Coefficients We will consider solutions of the form: Why ? There is a reason. We will discuss it later in the course.

Second Order Linear Homogenous Equations with Constant Coefficients We will consider solutions of the form:

Second Order Linear Homogenous Equations with Constant Coefficients We will consider solutions of the form: Then :

Second Order Linear Homogenous Equations with Constant Coefficients We will consider solutions of the form: Then :

Second Order Linear Homogenous Equations with Constant Coefficients We will consider solutions of the form: Then :

Second Order Linear Homogenous Equations with Constant Coefficients We will consider solutions of the form: Then :

Second Order Linear Homogenous Equations with Constant Coefficients Factoring out: One of these two must be 0

Second Order Linear Homogenous Equations with Constant Coefficients Factoring out: Can’t be 0

Second Order Linear Homogenous Equations with Constant Coefficients Factoring out: Must be 0

Second Order Linear Homogenous Equations with Constant Coefficients Factoring out: SoSo This is called the characteristic equation.

So to solve… Find the characteristic equation Find (Either factor or use the quadratic equation) Write the general solution:

The catch… Because there’s always a catch…. Find Two solutions! Three possibilities or

So to solve… Find the characteristic equation Find Write the general solution: and

Specific Solutions For first order equations, need 1 initial condition For second order equations, need 2 initial conditions For order N equations, need N initial conditions

Specific Solution For second order equations, need 2 initial conditions To find specific solution: Plug in Initial Conditions:

Specific Solution For second order equations, need 2 initial conditions To find specific solution: Plug in Initial Conditions: Solve for: and

So to solve… Find the characteristic equation Find Write the general solution: and Plug in Initial Conditions: Solve for: and

Example Find the characteristic equation Findand

Example Find the characteristic equation Findand Write the general solution: Plug in Initial Conditions: Solve for and

Example Find the characteristic equation Findand Write the general solution: Plug in Initial Conditions: Solve for and

Summary Higher Order Equations are Difficult To Solve Can solve Second Order Homogeneous Linear Equations with Constant Coefficients Using Characteristic Equations Need 2 Initial Conditions for Second Order Equations In general, need N initial conditions for Nth order equation

Questions?