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 $n$-th order ODE has the general form

$$y^{(n)}+f_1(x)y^{(n-1)}+\dots +f_{n-1}(x)y^{\prime }+f_n(x)y=F(x)$$  where $f_1(x)$,…,$f_n(x)$, and $F(x)$ are continuous real-valued functions on some interval $I=(a,b)$. When $F(x)=0$, we call this ODE homogeneous, otherwise the ODE is nonhomogeneous.  Notice for an $n$-th order ODE, there are typically $n$ initial conditions: $y(x_0)=y_0,$ $y^{\prime }(x_0)=y_1$,…,$y^{(n-1)}(x_0)=y_{n-1}$. Like before, we want to find out when an $n$-th order ODE has exactly one solution. The principal is actually the same!

Theorem: Consider the $n$-th order initial value problem:

$$\begin{array}{cc}{y}^{(n)}+f_1(x)y^{(n-1)}+\dots +f_{n-1}(x)y^{\prime }+f_n(x)y& =F(x)\\ y(x_0)=y_0,y^{\prime }(x_0)=y_1,\dots ,y^{(n-1)}(x_0)=y_{n-1}.\end{array}$$

 If $f_1(x)$,…,$f_n(x)$, and $F(x)$ are continuous on an open interval $I$, then there exists exactly one solution $y=g(x)$ that satisfies the initial value problem. 

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

Exercise 1:  $(x+2)y^{(4)}+(x-3)y^{″}+\text{cot}(x)y=0$. Determine intervals in which solutions are sure to exist. Suppose $y(4)=y^{\prime }(4)=y^{″}(4)=y^{‴}(4)=0$. Determine the largest interval such that there exists one unique solution.  Like the $2$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, $y_1,...,y_n$ is

$$\begin{array}{rl}W(y_1,y_2,\dots ,y_n)& =\left|\begin{array}{cccc}{y}_1& y_2& \dots & y_n\\ y_1^{\prime }& y_2^{\prime }& \dots & y_n^{\prime }\\ ⋮\\ y_1^{(n-1)}& y_2^{(n-1)}& \cdots & y_n^{(n-1)}\end{array}\right|.\end{array}$$

Example 2: Find the Wronskian of $y_1(x)=3$, $y_2(x)=x-3$, and $y_3(x)=x^2+4x$.

Exercise 2: Find the Wronskian of $y_1(x)=1$, $y_2(x)=x+2$, and $y_3(x)=4+3x$.

Theorem: Consider the $n$-th order homogeneous ODE: 

$$\begin{array}{rl}{y}^{(n)}+f_1(x)y^{(n-1)}+\dots +f_{n-1}(x)y^{\prime }+f_n(x)y& =0.\end{array}$$

 If $f_1(x)$,…,$f_n(x)$ are continuous on an open interval $I$, and $y_1$,…,$y_n$ are solutions of ODE with Wronskian, $W(y_1,...,y_n)(x)\ne 0$ for at least one $x$, then every solution $y$ of the ODE can be written as a linear combination of $y_1,...,y_n:$

$$y=c_1{y}_1+c_2{y}_2+\dots +c_n{y}_n.$$

The set $\{y_1,...,y_n\}$ is called a fundamental set of solutions. $y=c_1{y}_1+c_2{y}_2+...+c_n{y}_n$ 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. $y^{‴}+2y^{″}-y^{\prime }-2y=0$, and $y_1=e^t$, $y_2=e^{-t}$, and $y_3=e^{-2t}$.

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. $y^{‴}-4y^{\prime }=0$, and $y_1=e^{2t}$, $y_2=e^{-2t}$, and $y_3=5$. Similar to $2$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 $n$-th order nonhomogeneous ODE: 

$$\begin{array}{rl}{y}^{(n)}+f_1(x)y^{(n-1)}+\dots +f_{n-1}(x)y^{\prime }+f_n(x)y& =F(x).\end{array}$$

If $y_p(x)$ is a particular solution of the ODE, and $y_c=c_1{y}_1+c_2{y}_2+...+c_n{y}_n$ is a general solution of the homogeneous ODE, $y^{(n)}+f_1(x)y^{(n-1)}+...+f_{n-1}(x)y^{\prime }+f_n(x)y=0$, then the general solution of the nonhomogeneous ODE is 

$$y(x)=y_c+y_p.$$

Group Work: Find solutions.

1. $x^3-3x^2+4x=0$

2. $x^4-16=0$

3. $x^3+9x=0$

4. $x^3-6x^2+12x-8=0$

5. $x^4+8x^2+16=0$