Linear, Constant Coefficient, Homogeneous
Differential Equations

Steve Dunbar

Department of Mathematics and Statistics
University of Nebraska-Lincoln
Lincoln, NE 68588-0323
sdunbar@mathlab01.unl.edu

Fall Semester 1993

Covers Nagel & Saff, Sections 4.6, 6.2, 6.3

Before Starting

Define a new Mathematica function to compute the Wronskian of two functions.

Wronskian[f1_, f2_] := Det[{{f1,f2},{D[f1,x], D[f2,x]}}]

Fundamental Sets of Solutions

Use the Mathematica command DSolve to analytically solve second order differential equations.   
The solution returns as a solution rule.  
From the solution rule, we can read off the fundmental set of solutions.

DSolve[ 2*y''[x] + 9*y'[x] + 9*y[x] == 0, y[x],x]

From this, we can see that the fundamental set of solutions is Exp[-3 x] and Exp[-3 x/2]

Wronskian[ Exp[-3*x], Exp[-3*x/2] ]

We can also look at the auxiliary equation, and solve directly for the eignevalues:

Solve[ 2*r^2 + 9*r + 9 == 0, r]

Factor[2*r^2 + 9*r + 9 ]

Experiments to Perform

(Borelli and Coleman, Experiment 4.2.3 , page 122)  Consider the homogeneous, constant coeffiicient equation 
 	y''  - y' + b*y == 0.
 Solve  y(x) for  values of b = -1, 0, 1/4, 1 .  
 Find the fundametnal set of solutions in each case.
 Evaluate the Wronskian in each case.
 Find the eigenvalues by solving the auxiliary equation in each case.

(Borelli and Coleman, Experiment 4.2.4, page 122)  Consider the homogeneous, constant coefficient equation 
 	y''  - a*y'  - y == 0 for a = -1, 0, 1/4, 1.
 Find the fundametnal set of solutions in each case.
 Evaluate the Wronskian in each case.
 Find the eigenvalues by solving the auxiliary equation in each case.

Higher Order Equations

(Nagle & Saff, Section6.3, page 258, #8) Solve the  third-order, constant coefficient, homogeneous differential equation
	y''' +5 y'' -13 y' + 7 y = 0
with DSolve.

Find the eigenvalues by using the Solve command on the auxiliary equation.

Find the eigenvalues by using the Factor command on the auxiliary polynomial.

(Nagle & Saff, Section6.3, page 259, #25) Solve the  third-order, constant coefficient, homogeneous differential equation
	y''' +5 y'' -13 y' + 7 y = 0
with DSolve.

Find the eigenvalues by using the Solve command on the auxiliary equation.

Find the eigenvalues by using the Factor command on the auxiliary polynomial.  What happens?

Try to estimate the eigenvalues by graphing the auxiliary polynomial and observing the root crossings.