Differential Operators,  
 & Second Order Linear 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 Nagle & Saff, Section 4.2, 4.6

Experiments with Differential Operators

Define some Linear Constant Coefficient Differential Operators

L1[y] = y'' - 2y' + y

L1[f_] := D[f,{x,2}] - 2 D[f,x] + f

L2[y] = y''+4 y

L2[f_] := D[f,{x,2}] + 4 f

L3[y] = 2y'' + 9 y' + 9 y

L3[f_] := 2 D[f,{x,2}] + 9 D[f,x] + 9 f

Try some functions into these differential operators

Experiments with L1

L1[ x^3]

L1[ Sin[b x]]

L1[Cos[b x]]

L1[ Exp[a x]]

L1[ Exp[a x] Cos[b x] ]

Simplify[%]

Experinents with L2

L2[ x^3]

L2[ Cos[b x]]

Simplify[%]

Simplify[ L2[ Sin[b x]]]

Simplify[ L2[ Exp[ a x] Cos[ b x]]]

Experiments with L3

L3[ x^3]

Simplify[ L3[ Exp[ a x]]]

Simplify[ L3[ Exp[ a x] Cos[ b x]]]

Question to answer:

What conclusion can you make about the image of 
an exponential function, a sinusoidal function, a polynomial, or a product of these 
under a linear constant coefficient differential operator?

Experiments with Solutions of Second Order Linear 
Differential Equations

Example of solution and plotting

Use the Mathematica command NDSolve to numerically solve second order differnetial equations.  NDSolve uses a very advanced and sophisticated method,  similar in spirit to the improved Euler predictor-corrector method.

exsoln = NDSolve[{ 2*y''[x] + 9*y'[x] + 9*y[x] == 0, 
                   y[0] == 1, y'[0] == -1}, 
                   y[x], {x,0,10}]

Plot[ y[x] /. exsoln, {x, 0, 10}, PlotRange -> All]

Experiments to Perform

(Borelli and Coleman, Experiment 4.2.3 , page 122)  Consider the IVP 
 	y''  - y' + b*y == 0,	y(0) == 1, y'(0) == -1.
 Graph y(x) for several values of b, (positive, negative, and zero).  Are there any values of b for which the solution is oscillatory?  What do you guess is the general form of the solutions?

(Borelli and Coleman, Experiment 4.2.4, page 122)  Consider the IVP 
 	y''  - a*y'  - y == 0,	y(0) == 1, y'(0) == -1.
 Graph y(x) for several values of a, (positive, negative, and zero).  Are there any values of a for which the solution is oscillatory?  What do you guess is the general form of the solutions?

Extra Credit::  We suggested that the leading coefficient  in a second order equation can't be 0.  What happens to solutions if we try an equation with a leading coefficient nearly 0?

Consider the IVP 
 	epsilon y''  + y'  +  y == 0,	y(0) == 1, y'(0) == -1.
 Graph y(x) for several values of epsilon, (positive, negative, and zero).  
 Are there any values of epsilon for which the solution is oscillatory?  
 Does Mathematica make any complaints, or give any warning messages?