Section 3.2 Higher Order Differential equations

Objective:

1. Find the solution of a high order homogenous differential equations with constant coefficients.

Definition: Consider the [latex]n[/latex]-th order homogeneous ODE: 

\[\begin{align*} a_{0}y^{(n)}+a_{1}y^{(n-1)}+…+a_{n-1}y’+a_{n}y & =0 \end{align*}\]

 where [latex]a_{1},...,a_{n}[/latex] are real coefficients. The characteristic equation of the ODE [latex]a_{0}y^{(n)}+a_{1}y^{(n-1)}+...+a_{n}y=0[/latex] is [latex]a_{0}\lambda^{n}+a_{1}\lambda^{n-1}+...+a_{n}=0[/latex].

Fact: If [latex]r[/latex] is a root of the characteristic equation of the ODE with multiplicity [latex]p[/latex] then [latex]e^{rt},te^{rt},..,t^{p-1}e^{rt}[/latex] are solutions of the ODE.

Example 1:  Find the general solution of the ODE. [latex]y^{'''}-3y''-4y'=0[/latex]

Exercise 1: Find the general solution of the ODE. [latex]y^{'''}-2y''-8y'=0[/latex]

Example 2: Find the general solution of the ODE. [latex]y^{(4)}-81y=0[/latex]

Exercise 2: Find the general solution of the ODE. [latex]y^{(4)}-625y=0[/latex]

Example 3: Find the general solution of the ODE. [latex]y'''+9y'=0[/latex]

Exercise 3: Find the general solution of the ODE. [latex]y'''+4y'=0[/latex]

Example 4: Find the general solution of the ODE. [latex]y'''-6y''+12y'-8y=0[/latex]

Exercise 4: Find the general solution of the ODE. [latex]y'''+3y''+3y'+y=0[/latex]

Group Work: 

1. [latex]y^{'''}-2y''+2y'=0[/latex] and [latex]y(0)=1[/latex], [latex]y'(0)=0[/latex], [latex]y''(0)=-1[/latex]. Find the solution. 

2. [latex]y^{'''}+y'=0[/latex] and [latex]y(0)=2[/latex], [latex]y'(0)=1[/latex], [latex]y''(0)=0[/latex]. Find the solution. 

3. [latex]y^{(4)}-4y'''+4y''=0[/latex] and [latex]y(0)=2[/latex], [latex]y'(0)=1[/latex], [latex]y''(0)=0,[/latex] [latex]y'''(0)=0[/latex]. Find the solution.