Homogeneous Second Order Ordinary Differential Equations
with Constant Coefficients
Table of Contents
1. Introduction
2. The Characteristic Equation
3. Case 1: Distinct Real Roots
4. Case 2: Repeated Root
5. Case 3: Complex Conjugate Roots
6. The Wronskian and Linear Independence
7. Initial Value Problems vs Boundary Value Problems
8. Behavior Analysis
9. Summary Table
10. Common Mistakes
1. Introduction
A second-order linear ordinary differential equation is an equation of the form:
a (x) y ′ ′ +b (x) y ′ + c( x ) y=g( x )
where a(x), b(x), c(x), and g(x) are continuous functions on some interval. When g(x) = 0, the
equation is called homogeneous. When the coefficients a(x), b(x), and c(x) are constants (not
functions of x), we have a homogeneous linear ODE with constant coefficients.
Standard Form
The standard form is written as:
ay ′ ′+ by ′+ cy=0
where a, b, and c are constants with a ≠ 0. We often normalize by dividing by a to get:
y ′ ′+ py ′ +qy =0
This type of equation appears frequently in physics and engineering, modeling systems like
spring-mass systems, RLC electrical circuits, and damped oscillations.
, 2. The Characteristic Equation
The key to solving homogeneous linear ODEs with constant coefficients is the characteristic
equation, which comes from assuming a trial solution of the form:
rx
y=e e
where r is a constant to be determined. Let us derive the characteristic equation.
Derivation
Given y = e^(rx), we compute the derivatives:
rx
y ′=ℜ e
rx
y ′ ′=r ² e e
Substituting these into the differential equation ay″ + by′ + cy = 0:
rx rx rx
ar ² e e +bre e +ce e =0
Since e^(rx) ≠ 0 for all x, we can factor it out:
rx
e e (ar ²+ br+ c)=0
This gives us the characteristic equation (also called the auxiliary equation):
ar ²+ br +c=0
The Discriminant
The nature of the roots of the characteristic equation depends on the discriminant:
Δ=b ²−4 ac
The sign of Δ determines whether we have distinct real roots, a repeated real root, or complex
conjugate roots. This leads to three cases:
3. Case 1: Distinct Real Roots (Δ > 0)
When the discriminant Δ > 0, the characteristic equation ar² + br + c = 0 has two distinct real
roots:
−b ± √(b ²−4 ac )
r 1 ,21 , 2=
2a
General Solution
Since y₁ = e^(r₁x) and y₂ = e^(r₂x) are two linearly independent solutions (which can be
verified using the Wronskian), the general solution is:
r₁ x r₂x
y ( x )=c 11 e e + c 22 e e
where c₁ and c₂ are arbitrary constants determined by initial conditions.
with Constant Coefficients
Table of Contents
1. Introduction
2. The Characteristic Equation
3. Case 1: Distinct Real Roots
4. Case 2: Repeated Root
5. Case 3: Complex Conjugate Roots
6. The Wronskian and Linear Independence
7. Initial Value Problems vs Boundary Value Problems
8. Behavior Analysis
9. Summary Table
10. Common Mistakes
1. Introduction
A second-order linear ordinary differential equation is an equation of the form:
a (x) y ′ ′ +b (x) y ′ + c( x ) y=g( x )
where a(x), b(x), c(x), and g(x) are continuous functions on some interval. When g(x) = 0, the
equation is called homogeneous. When the coefficients a(x), b(x), and c(x) are constants (not
functions of x), we have a homogeneous linear ODE with constant coefficients.
Standard Form
The standard form is written as:
ay ′ ′+ by ′+ cy=0
where a, b, and c are constants with a ≠ 0. We often normalize by dividing by a to get:
y ′ ′+ py ′ +qy =0
This type of equation appears frequently in physics and engineering, modeling systems like
spring-mass systems, RLC electrical circuits, and damped oscillations.
, 2. The Characteristic Equation
The key to solving homogeneous linear ODEs with constant coefficients is the characteristic
equation, which comes from assuming a trial solution of the form:
rx
y=e e
where r is a constant to be determined. Let us derive the characteristic equation.
Derivation
Given y = e^(rx), we compute the derivatives:
rx
y ′=ℜ e
rx
y ′ ′=r ² e e
Substituting these into the differential equation ay″ + by′ + cy = 0:
rx rx rx
ar ² e e +bre e +ce e =0
Since e^(rx) ≠ 0 for all x, we can factor it out:
rx
e e (ar ²+ br+ c)=0
This gives us the characteristic equation (also called the auxiliary equation):
ar ²+ br +c=0
The Discriminant
The nature of the roots of the characteristic equation depends on the discriminant:
Δ=b ²−4 ac
The sign of Δ determines whether we have distinct real roots, a repeated real root, or complex
conjugate roots. This leads to three cases:
3. Case 1: Distinct Real Roots (Δ > 0)
When the discriminant Δ > 0, the characteristic equation ar² + br + c = 0 has two distinct real
roots:
−b ± √(b ²−4 ac )
r 1 ,21 , 2=
2a
General Solution
Since y₁ = e^(r₁x) and y₂ = e^(r₂x) are two linearly independent solutions (which can be
verified using the Wronskian), the general solution is:
r₁ x r₂x
y ( x )=c 11 e e + c 22 e e
where c₁ and c₂ are arbitrary constants determined by initial conditions.
