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 $n$-th order homogeneous ODE: 

$$\begin{array}{rl}{a}_0{y}^{(n)}+a_1{y}^{(n-1)}+\dots +a_{n-1}{y}^{\prime }+a_{n}y& =0\end{array}$$

 where $a_1,...,a_n$ are real coefficients. The characteristic equation of the ODE $a_0{y}^{(n)}+a_1{y}^{(n-1)}+...+a_{n}y=0$ is $a_0{\lambda }^n+a_1{\lambda }^{n-1}+...+a_n=0$.

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

Example 1:  Find the general solution of the ODE. $y^{{}^{‴}}-3y^{″}-4y^{\prime }=0$

Exercise 1: Find the general solution of the ODE. $y^{{}^{‴}}-2y^{″}-8y^{\prime }=0$

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

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

Example 3: Find the general solution of the ODE. $y^{‴}+9y^{\prime }=0$

Exercise 3: Find the general solution of the ODE. $y^{‴}+4y^{\prime }=0$

Example 4: Find the general solution of the ODE. $y^{‴}-6y^{″}+12y^{\prime }-8y=0$

Exercise 4: Find the general solution of the ODE. $y^{‴}+3y^{″}+3y^{\prime }+y=0$

Group Work: 

1. $y^{{}^{‴}}-2y^{″}+2y^{\prime }=0$ and $y(0)=1$, $y^{\prime }(0)=0$, $y^{″}(0)=-1$. Find the solution. 

2. $y^{{}^{‴}}+y^{\prime }=0$ and $y(0)=2$, $y^{\prime }(0)=1$, $y^{″}(0)=0$. Find the solution. 

3. $y^{(4)}-4y^{‴}+4y^{″}=0$ and $y(0)=2$, $y^{\prime }(0)=1$, $y^{″}(0)=0,$ $y^{‴}(0)=0$. Find the solution.