Section 3.1 Higher Order Differential equations

Objective:

1. Definition of a higher order differential equation

2. Uniqueness of the solution

3. Wronskian of higher order

Definition: An [latex]n[/latex]-th order ODE has the general form

\[y^{(n)}+f_{1}(x)y^{(n-1)}+…+f_{n-1}(x)y’+f_{n}(x)y=F(x)\]  where [latex]f_{1}(x)[/latex],…,[latex]f_{n}(x)[/latex], and [latex]F(x)[/latex] are continuous real-valued functions on some interval [latex]I=(a,b)[/latex]. When [latex]F(x)=0[/latex], we call this ODE homogeneous, otherwise the ODE is nonhomogeneous.  Notice for an [latex]n[/latex]-th order ODE, there are typically [latex]n[/latex] initial conditions: [latex]y(x_{0})=y_{0},[/latex] [latex]y'(x_{0})=y_{1}[/latex],…,[latex]y^{(n-1)}(x_{0})=y_{n-1}[/latex]. Like before, we want to find out when an [latex]n[/latex]-th order ODE has exactly one solution. The principal is actually the same!

Theorem: Consider the [latex]n[/latex]-th order initial value problem:

$$\begin{array}{cccc} y^{(n)}+f_{1}(x)y^{(n-1)}+…+f_{n-1}(x)y’+f_{n}(x)y & =F(x)\\y(x_{0})=y_{0},y'(x_{0})=y_{1},…,y^{(n-1)}(x_{0})=y_{n-1}.\end{array}$$

 If [latex]f_{1}(x)[/latex],…,[latex]f_{n}(x)[/latex], and [latex]F(x)[/latex] are continuous on an open interval [latex]I[/latex], then there exists exactly one solution [latex]y=g(x)[/latex] that satisfies the initial value problem. 

Example 1: [latex](x-1)y^{(4)}+(x+2)y''+\text{tan}(x)y=0[/latex]. Determine intervals in which solutions are sure to exist. Suppose [latex]y(3)=y'(3)=y''(3)=y'''(3)=7[/latex]. Determine the largest interval such that there exists one unique solution.

Exercise 1:  [latex](x+2)y^{(4)}+(x-3)y''+\text{cot}(x)y=0[/latex]. Determine intervals in which solutions are sure to exist. Suppose [latex]y(4)=y'(4)=y''(4)=y'''(4)=0[/latex]. Determine the largest interval such that there exists one unique solution.  Like the [latex]2[/latex]nd order ODE, Wronskian plays an important role on deciding if a set of solution is a fundamental set of solution. We define the Wronskian for a set of solutions. 

Definition: The Wronskian of a set of solution, [latex]y_{1},...,y_{n}[/latex] is

$$\begin{align*} W(y_{1},y_{2},…,y_{n}) & =\left|\begin{array}{cccc} y_{1} & y_{2} & … & y_{n}\\y_{1}’ & y_{2}’ & … & y_{n}’\\ \vdots\\y_{1}^{(n-1)} & y_{2}^{(n-1)} & \cdots & y_{n}^{(n-1)} \end{array}\right|.\end{align*}$$

Example 2: Find the Wronskian of [latex]y_{1}(x)=3[/latex], [latex]y_{2}(x)=x-3[/latex], and [latex]y_{3}(x)=x^{2}+4x[/latex].

Exercise 2: Find the Wronskian of [latex]y_{1}(x)=1[/latex], [latex]y_{2}(x)=x+2[/latex], and [latex]y_{3}(x)=4+3x[/latex].

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

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

 If [latex]f_{1}(x)[/latex],…,[latex]f_{n}(x)[/latex] are continuous on an open interval [latex]I[/latex], and [latex]y_{1}[/latex],…,[latex]y_{n}[/latex] are solutions of ODE with Wronskian, [latex]W(y_{1},...,y_{n})(x)\neq0[/latex] for at least one [latex]x[/latex], then every solution [latex]y[/latex] of the ODE can be written as a linear combination of [latex]y_{1},...,y_{n}:[/latex]

\[y=c_{1}y_{1}+c_{2}y_{2}+…+c_{n}y_{n}.\]

The set [latex]\{y_{1},...,y_{n}\}[/latex] is called a fundamental set of solutions. [latex]y=c_{1}y_{1}+c_{2}y_{2}+...+c_{n}y_{n}[/latex] is called the general solution of the ODE.

Example 3: Verify the given functions are solutions of the ODE and determine if they form a fundament set of the solutions and find the general solution of the ODE. [latex]y'''+2y''-y'-2y=0[/latex], and [latex]y_{1}=e^{t}[/latex], [latex]y_{2}=e^{-t}[/latex], and [latex]y_{3}=e^{-2t}[/latex].

Exercise 3: Verify the given functions are solutions of the ODE and determine if they form a fundament set of the solutions and find the general solution of the ODE. [latex]y'''-4y'=0[/latex], and [latex]y_{1}=e^{2t}[/latex], [latex]y_{2}=e^{-2t}[/latex], and [latex]y_{3}=5[/latex]. Similar to [latex]2[/latex]nd order ODE, the solution set of the nonhomogeneous ODE is the sum of the general solution of the homogeneous part and a particular solution. 

Fact:  Consider the [latex]n[/latex]-th order nonhomogeneous ODE: 

$$\begin{align*} y^{(n)}+f_{1}(x)y^{(n-1)}+…+f_{n-1}(x)y’+f_{n}(x)y & =F(x). \end{align*}$$

If [latex]y_{p}(x)[/latex] is a particular solution of the ODE, and [latex]y_{c}=c_{1}y_{1}+c_{2}y_{2}+...+c_{n}y_{n}[/latex] is a general solution of the homogeneous ODE, [latex]y^{(n)}+f_{1}(x)y^{(n-1)}+...+f_{n-1}(x)y'+f_{n}(x)y=0[/latex], then the general solution of the nonhomogeneous ODE is 

\[y(x)=y_{c}+y_{p}.\]

Group Work: Find solutions.

1. [latex]x^{3}-3x^{2}+4x=0[/latex]

2. [latex]x^{4}-16=0[/latex]

3. [latex]x^{3}+9x=0[/latex]

4. [latex]x^{3}-6x^{2}+12x-8=0[/latex]

5. [latex]x^{4}+8x^{2}+16=0[/latex]