BSU Math 333 (Ultman)
Worksheet: Second Order Linear Homogenous Differential
Equations With Constant Coefficents
Prerequisites
In order to learn the new skills and ideas presented in this worksheet, you must:
Be able to find the general solution to a second order linear differential equation with constant
coefficients, by finding the roots of the characteristic polynomial.
Be able to convert a second order differential equation into a system of first order equations.
Be able to solve a system of first order linear homogenous differential equations with constant
coefficents.
Goals
In this worksheet, you will:
Find the particular solution to to a second order linear differential equation with constant
coefficients, subject to initial conditions.
Compare the general solution of a second order linear differential equation with constant coefficients, obtained in two different ways: by finding the roots of the characteristic polynomial,
and by solving the associated system of first order equations.
BSU Math 333 (Ultman)
Worksheet: Solutions to Second Order Linear Differential Equations 1
Second Order Linear Homogenous Constant Coefficient Initial Value
Problems
A mass of m = 2 kg is suspended from a spring
with spring constant k = 4 (units in kg/s2 ), and
subject to a damping force with coefficient γ = 6
(units in kg/s). The mass is pulled down 0.1
m from equilibrium position, and released. The
system is not subject to any external forces.
This mass-spring system is modeled by the differential equation:
my 00 + γy 0 + ky = 0
1. Using one sentence for each, describe what the functions y(t), y 0 (t), and y 00 (t) represent.
2. What are the initial conditions for this problem? State them mathematically, and describe in
a sentence what they represent.
3. Set up the initial value problem (that is, fill in the given values for the coefficients m, γ, and
k, and write the initial conditions).
4. Find the general solution to the differential equation.
continued. . .
BSU Math 333 (Ultman)
Worksheet: Solutions to Second Order Linear Differential Equations 2
5. Now, you are going to find the particular solution that satisfies the given initial conditions.
To do this:
(a) Take the derivative of the general solution you found in the previous question.
(b) Use the initial conditions, together with the general solution and its derivative, to create
a system of two equations in c1 and c2 .
(c) Solve the system from the previous equation to find the values for c1 and c2 .
(d) Write the solution to the initial value problem.
BSU Math 333 (Ultman)
Worksheet: Solutions to Second Order Linear Differential Equations 3
General Solutions to Second Order Linear Homogenous Equations With
Constant Coefficients: Comparing Characteristic Polynomial Method
and Systems of First Order Equations
Consider the differential equation:
y 00 + y 0 − 6y = 0.
1. Find the general solution to the equation using the characteristic polynomial.
2. Convert the equation to a system of first order equations, by assigning the variables x1 = y,
x2 = y 0 , and solve the system using the eigenvalue/eigenvector method.
continued. . .
BSU Math 333 (Ultman)
Worksheet: Solutions to Second Order Linear Differential Equations 4
3. Compare the characteristic polynomial and its roots from (1) with det A − λI = 0 and the
eigenvalues from (2). What do you notice?
4. Compare the general solution to the equation from (1) with the solution to the system from
(2). What do you notice?
5. Take the derivative of the general solution from (1), and compare it to the solution to the
system from (2). What do you notice?