Unit III: Higher-Order Linear Differential Equations

1. Introduction to Homogeneous and Nonhomogeneous LDEs

A linear differential equation (LDE) of order n (where n > 1) contains the dependent variable y and its derivatives up to order n, all raised to the first power and not multiplied together.

a_n(x)y⁽ⁿ⁾ + a_n-1(x)y^(n-1) + ... + a₁(x)y' + a₀(x)y = g(x)

Here, a_n(x), a_n-1(x), ..., a₀(x) are coefficients (which can be functions of x or constants), and g(x) is the input, driving, or forcing term. The general nature of the equation depends entirely on whether g(x) is zero or non-zero:

2. Linear Dependence, Independence, and the Wronskian

Determining whether a set of functions is linearly independent is necessary for building the general solution of a differential equation.

Linear Dependence and Independence

A set of functions f₁(x), f₂(x), ..., f_n(x) is linearly dependent on an interval I if there exist constants c₁, c₂, ..., c_n (not all zero) such that:

c₁f₁(x) + c₂f₂(x) + ... + c_nf_n(x) = 0

for every x in I. If the only set of constants that satisfies this equation is c₁ = c₂ = ... = c_n = 0, then the set of functions is linearly independent.

The Wronskian

The Wronskian is a mathematical tool used to test the linear independence of differentiable functions. For n functions, the Wronskian is defined as the determinant of an n × n matrix containing the functions in the first row and their successive derivatives in the subsequent rows.

W(f₁, f₂, ..., f_n) = det | Row 1: f₁, f₂, ..., f_n | Row 2: f₁', f₂', ..., f_n' | ... | Row n: f₁^(n-1), f₂^(n-1), ..., f_n^(n-1) |

For two functions, y₁ and y₂, the Wronskian is calculated as:

W(y₁, y₂) = y₁y₂' - y₂y₁'

For three functions, y₁, y₂, and y₃, the Wronskian is computed using a 3 × 3 determinant:

W(y₁, y₂, y₃) = y₁(y₂'y₃'' - y₃'y₂'') - y₂(y₁'y₃'' - y₃'y₁'') + y₃(y₁'y₂'' - y₂'y₁'')

Wronskian Test for Solutions

Theorem: Let y₁, y₂, ..., y_n be n solutions of an n-th order homogeneous linear differential equation on an interval I. The solutions are linearly independent on I if and only if their Wronskian is non-zero for all x in the interval:
W(y₁, y₂, ..., y_n) ≠ 0

Step-by-Step Example: Determine if the functions y₁ = e^2x and y₂ = e^-3x are linearly independent.

1. Find the first derivatives: y₁' = 2e^2x and y₂' = -3e^-3x.
2. Set up the Wronskian determinant: W = y₁y₂' - y₂y₁'.
3. Calculate: W = e^2x(-3e^-3x) - e^-3x(2e^2x) = -3e^-x - 2e^-x = -5e^-x.
4. Since -5e^-x is never zero for any real x, the functions are linearly independent.

Common Mistake: Confusing arbitrary functions with solutions to a LDE. For arbitrary functions, W(x) = 0 at some points does not automatically mean they are dependent. However, for actual solutions of a homogeneous LDE, the Wronskian is either identically zero (linearly dependent) or never zero (linearly independent) on the entire interval.

3. Basic Theory of Linear Differential Equations

The study of higher-order linear differential equations is grounded in three main theoretical principles: existence/uniqueness, superposition, and the structure of general solutions.

Existence and Uniqueness Theorem

Let the coefficients a_n(x), a_n-1(x), ..., a₀(x) and the function g(x) be continuous on an interval I, with a_n(x) ≠ 0 for all x in I. If x₀ is any point in I, then there exists a unique solution y(x) of the initial-value problem satisfying:
y(x₀) = y₀, y'(x₀) = y₁, ..., y^(n-1)(x₀) = y_n-1

The Superposition Principle

For homogeneous equations, any linear combination of solutions is also a solution.

If y₁, y₂, ..., y_k are solutions of an n-th order homogeneous linear differential equation on an interval I, then the linear combination:
y = c₁y₁ + c₂y₂ + ... + c_ky_k
is also a solution on the interval, where c₁, c₂, ..., c_k are arbitrary constants.

Fundamental Set of Solutions

Any set of n linearly independent solutions y₁, y₂, ..., y_n of an n-th order homogeneous linear differential equation is called a fundamental set of solutions. This set forms the basis for the entire solution space.

General Solution of a Homogeneous Equation

If y₁, y₂, ..., y_n is a fundamental set of solutions, then the general solution is expressed as:

y = c₁y₁ + c₂y₂ + ... + c_ny_n

General Solution of a Nonhomogeneous Equation

The general solution of a nonhomogeneous linear differential equation is the sum of the complementary function and a particular solution:

y = y_c + y_p

Where:

• y_c (Complementary Function): The general solution of the associated homogeneous equation (y_c = c₁y₁ + ... + c_ny_n).
• y_p (Particular Solution): Any specific, constant-free solution that satisfies the nonhomogeneous equation.

4. Higher-Order LDEs with Constant Coefficients (up to 4th Order)

When the coefficients a_i are constants, the homogeneous differential equation can be solved algebraically. Consider the equation:

a_ny⁽ⁿ⁾ + a_n-1y^(n-1) + ... + a₁y' + a₀y = 0

Substituting the trial solution y = e^rx produces the Auxiliary (Characteristic) Equation:

a_nrⁿ + a_n-1r^n-1 + ... + a₁r + a₀ = 0

The roots of this auxiliary polynomial determine the form of the general solution, categorized into four cases:

Case-by-Case Analysis of Roots

Step-by-Step Solved Examples

Example 1 (2nd Order - Distinct Real Roots): Solve y'' - 5y' + 6y = 0.

1. Write the auxiliary equation: r² - 5r + 6 = 0.
2. Factor the quadratic equation: (r - 2)(r - 3) = 0.
3. Identify the roots: r₁ = 2, r₂ = 3.
4. Formulate the general solution:

   y = c₁e^2x + c₂e^3x

Example 2 (3rd Order - Repeated Real Roots): Solve y''' - 3y'' + 3y' - y = 0.

1. Write the auxiliary equation: r³ - 3r² + 3r - 1 = 0.
2. Recognize the perfect cube expansion: (r - 1)³ = 0.
3. Identify the roots: r = 1 with a multiplicity of 3.
4. Formulate the general solution using x-multipliers:

   y = (c₁ + c₂x + c₃x²)e^x

Example 3 (4th Order - Mixed Distinct Real and Complex Roots): Solve y⁽⁴⁾ - 16y = 0.

1. Write the auxiliary equation: r⁴ - 16 = 0.
2. Factor using the difference of squares: (r² - 4)(r² + 4) = 0.
3. Factor completely: (r - 2)(r + 2)(r² + 4) = 0.
4. Determine the roots:
   • Real roots: r = 2, -2
   • Complex roots: r = ±2i (where α = 0, β = 2)
5. Write the general solution:

   y = c₁e^2x + c₂e^-2x + c₃cos(2x) + c₄sin(2x)

Example 4 (4th Order - Repeated Real Roots): Solve y⁽⁴⁾ - 2y'' + y = 0.

1. Write the auxiliary equation: r⁴ - 2r² + 1 = 0.
2. Factor as a perfect square trinomial: (r² - 1)² = 0.
3. Factor further inside the square: [(r - 1)(r + 1)]² = (r - 1)²(r + 1)² = 0.
4. Identify the roots: r = 1 (multiplicity 2) and r = -1 (multiplicity 2).
5. Write the general solution:

   y = (c₁ + c₂x)e^x + (c₃ + c₄x)e^-x

Example 5 (4th Order - Repeated Complex Roots): Solve y⁽⁴⁾ + 8y'' + 16y = 0.

1. Write the auxiliary equation: r⁴ + 8r² + 16 = 0.
2. Factor the quadratic form: (r² + 4)² = 0.
3. Solve for r: r² = -4, which yields r = ±2i with a multiplicity of 2 (α = 0, β = 2).
4. Apply the repeated complex roots formula:

   y = (c₁ + c₂x)cos(2x) + (c₃ + c₄x)sin(2x)

Common Student Errors & Important Observations

• Constant Match Rule: An n-th order differential equation must always contain exactly n arbitrary constants (c₁, c₂, ..., c_n) in its general solution. If you are solving a 4th-order equation, double-check that you have 4 independent constants in your final answer.
• Forgetting the Independent Variable Multiplier: For repeated roots, never write c₁e^rx + c₂e^rx. The terms must be distinct: c₁e^rx + c₂xe^rx + c₃x²e^rx.
• Trigonometric Term Arguments: When writing the complex conjugate solution e^αx(c₁cos(βx) + c₂sin(βx)), do not write the imaginary unit i inside the sine or cosine functions. Also, ensure β is kept positive inside the trigonometric functions.