ODE-04

Higher-Order Methods

Higher-Order Linear Equations

Learn powerful techniques for reducing the order of differential equations, understanding first integrals, analyzing autonomous systems, and converting higher-order equations to first-order systems.

Learning Objectives

Apply reduction of order for missing variables
Identify and use first integrals
Solve autonomous (stationary) equations
Perform phase plane analysis
Convert higher-order ODEs to systems
Understand the structure of solution spaces
Solve physical systems using energy methods
Apply these techniques to tracking problems

1. Reduction of Order

For higher-order equations, reducing the order can significantly simplify the problem. The key insight is that when certain variables are missing from the equation, we can introduce substitutions that lower the order.

Definition 4.1: Order Reduction for Missing Variables

Consider the $n$ -th order equation:

F(t, x, x', \ldots, x^{(n)}) = 0

Case 1: If the equation has the form $F(t, x^{(k)}, \ldots, x^{(n)}) = 0$ (missing $x, x', \ldots, x^,{(k-1)}$ ), substitute $y = x^,{(k)}$ to get an $(n-k)$ -th order equation.

Case 2: If the equation has the form $F(x, x', \ldots, x^{(n)}) = 0$ (missing $t$ ), it is called autonomous. Substitute $y = x'$ and use $x'' = y\frac{dy}{dx}$ .

Theorem 4.1: Reduction for Missing Independent Variable

If $t$ does not appear explicitly in $F(x, x', \ldots, x^{(n)}) = 0$ , then setting $y = x'$ and expressing higher derivatives as:

x'' = y\frac{dy}{dx}, \quad x''' = y\frac{d}{dx}\left(y\frac{dy}{dx}\right), \quad \ldots

reduces the $n$ -th order equation to an $(n-1)$ -th order equation in $y$ as a function of $x$ .

Proof of Theorem 4.1:

Since $y = x' = dx/dt$ , by the chain rule:

x'' = \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} = y\frac{dy}{dx}

For the third derivative:

x''' = \frac{d}{dt}\left(y\frac{dy}{dx}\right) = \frac{d}{dx}\left(y\frac{dy}{dx}\right) \cdot \frac{dx}{dt} = y\frac{d}{dx}\left(y\frac{dy}{dx}\right)

By induction, each $x^,{(k)}$ can be expressed in terms of $y$ and its derivatives with respect to $x$ . ∎

Algorithm 4.1: Reduction of Order Method

Input: Higher-order ODE $F(t, x, x', \ldots, x^{(n)}) = 0$

Output: General solution

1. Check which variables are missing from the equation

2. If $x, x', \ldots, x^,{(k-1)}$ are missing:

2.1. Set $y = x^,{(k)}$

2.2. Solve the reduced equation for $y(t)$

2.3. Integrate $k$ times to find $x(t)$

3. If $t$ is missing (autonomous):

3.1. Set $y = x'$ , express $x'' = y \cdot dy/dx$

3.2. Solve for $y(x)$

3.3. Integrate $dx/dt = y(x)$ to find $x(t)$

4. return General solution with appropriate constants

Example 4.1: Equation Missing Lower Derivatives

Problem: Solve $t^2 x'' + tx' = 1$ .

Solution:

The equation is missing $x$ . Set $y = x'$ , so $y' = x''$ :

t^2 y' + ty = 1

This is a first-order linear equation. Dividing by $t^2$ :

y' + \frac{1}{t}y = \frac{1}{t^2}

The integrating factor is $\mu = e^{\int (1/t)dt} = t$ . Multiplying:

(ty)' = \frac{1}{t} \implies ty = \ln|t| + C_1 \implies y = \frac{\ln|t| + C_1}{t}

Integrating to find $x$ :

x = \int \frac{\ln|t| + C_1}{t}\,dt = \frac{1}{2}(\ln|t|)^2 + C_1\ln|t| + C_2

2. Autonomous Equations

Definition 4.2: Autonomous Equation

A differential equation is called autonomous (or stationary) if the independent variable $t$ does not appear explicitly:

F(x, x', x'', \ldots, x^{(n)}) = 0

The term "autonomous" comes from the fact that the system's behavior is independent of the absolute time—only the state matters.

Example 4.2: Physical System: Newton's Second Law

Problem: Solve $mx'' = f(x)$ where $m$ is mass and $f(x)$ is a position-dependent force.

Solution:

This is autonomous (no explicit $t$ ). Set $p = x'$ (momentum per unit mass):

x'' = \frac{dp}{dt} = \frac{dp}{dx}\frac{dx}{dt} = p\frac{dp}{dx}

The equation becomes:

mp\frac{dp}{dx} = f(x) \implies m\,p\,dp = f(x)\,dx

Integrating both sides:

\frac{1}{2}mp^2 = \int f(x)\,dx + C

This is the energy integral: $\frac{1}{2}m(x')^2 - \int f(x)dx = C$ , expressing conservation of mechanical energy (kinetic + potential = constant).

Remark:

The energy integral is a first integral of the equation. For conservative mechanical systems, multiplying the equation of motion by $x'$ and integrating always yields the energy conservation law.

3. First Integrals

Definition 4.3: First Integral

For the system $\frac{dx}{dt} = f(t, x)$ where $x \in \mathbb{R}^n$ , a function $\Phi(t, x) \in C^1(D)$ is called a first integral if:

$\Phi$ is not constant on $D$
$\Phi$ is constant along every solution curve in $D$

Equivalently, $\frac{d\Phi}{dt} = \frac{\partial \Phi}{\partial t} + \nabla_x \Phi \cdot f = 0$ along solutions.

Definition 4.4: Independent First Integrals

First integrals $\Phi_1, \ldots, \Phi_n$ are called independent if their Jacobian with respect to the state variables is non-zero:

\frac{\partial(\Phi_1, \ldots, \Phi_n)}{\partial(x_1, \ldots, x_n)} = \begin{vmatrix} \frac{\partial \Phi_1}{\partial x_1} & \cdots & \frac{\partial \Phi_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial \Phi_n}{\partial x_1} & \cdots & \frac{\partial \Phi_n}{\partial x_n} \end{vmatrix} \neq 0

Theorem 4.2: Existence and Uniqueness of First Integrals

If $f \in C^1(D)$ where $D \subset \mathbb{R}^{n+1}$ , then in a neighborhood of any regular point, the system $\frac{dx}{dt} = f(t, x)$ has exactly $n$ independent first integrals.

Proof of Theorem 4.2:

Existence: Consider the initial value problem with $x(t_0) = C = (C_1, \ldots, C_n)^T$ . Let $\Phi(t, C)$ denote the solution. By the implicit function theorem, we can locally solve $C_j = \Psi_j(t, x)$ for each $j$ . These $\Psi_j$ are first integrals.

Uniqueness (at most $n$ ): Taking the total derivative of any first integral:

\frac{d\Psi_j}{dt} = \frac{\partial \Psi_j}{\partial t} + \sum_{k=1}^{n} \frac{\partial \Psi_j}{\partial x_k} f_k = 0

For $n+1$ first integrals, the vector $(1, f_1, \ldots, f_n)$ would be a non-trivial solution to a homogeneous system, implying the Jacobian determinant is zero—contradicting independence. ∎

Theorem 4.3: Finding General Solution from First Integrals

If $\Phi_1(t, x) = C_1, \ldots, \Phi_n(t, x) = C_n$ are $n$ independent first integrals of $\frac{dx}{dt} = f(t, x)$ , then the general solution can be obtained by solving this system of algebraic equations for $x$ in terms of $t$ and the constants $C_1, \ldots, C_n$ .

Example 4.3: Finding First Integrals

Problem: Find first integrals for the system $x' = -y$ , $y' = x$ .

Solution:

This system describes rotation. Multiply the first equation by $x$ and the second by $y$ :

x\frac{dx}{dt} + y\frac{dy}{dt} = -xy + xy = 0

This gives:

\frac{1}{2}\frac{d}{dt}(x^2 + y^2) = 0 \implies x^2 + y^2 = C_1

For the second first integral, from $y' = x$ we have $x = y'$ . Substituting:

\frac{dy}{dt} = x \implies \frac{dy}{\sqrt{C_1 - y^2}} = dt

Integrating: $\arcsin(y/\sqrt{C_1}) = t + C_2$ , giving the second first integral.

4. Converting Higher-Order Equations to Systems

Theorem 4.4: Conversion to First-Order System

Any explicit $n$ -th order ODE $x^{(n)} = f(t, x, x', \ldots, x^{(n-1)})$ can be converted to a first-order system of dimension $n$ by setting:

x_1 = x, \quad x_2 = x', \quad \ldots, \quad x_n = x^{(n-1)}

The resulting system is:

\begin{cases} x_1' = x_2 \\ x_2' = x_3 \\ \vdots \\ x_{n-1}' = x_n \\ x_n' = f(t, x_1, x_2, \ldots, x_n) \end{cases}

Example 4.4: Converting a Second-Order Equation

Problem: Convert $x'' + 3x' + 2x = \sin t$ to a first-order system.

Solution:

Set $x_1 = x$ and $x_2 = x'$ . Then:

\begin{cases} x_1' = x_2 \\ x_2' = -2x_1 - 3x_2 + \sin t \end{cases}

In matrix form: $\mathbf{x}' = A\mathbf{x} + \mathbf{b}(t)$ where:

A = \begin{pmatrix} 0 & 1 \\ -2 & -3 \end{pmatrix}, \quad \mathbf{b}(t) = \begin{pmatrix} 0 \\ \sin t \end{pmatrix}

Remark:

Converting to a first-order system is fundamental because: (1) it unifies the theory—all results for first-order systems apply, (2) numerical methods are typically designed for first-order systems, and (3) the geometric interpretation (phase space) becomes clearer.

5. Applications: Missile Tracking Problem

Example 4.5: Missile Tracking Problem

Problem: A missile at the origin tracks a ship at $(0, H)$ moving east at speed $v_e$ . The missile travels at speed $v_w$ and always points toward the ship. Find the trajectory and interception point.

Solution:

Let $P(x(t), y(t))$ be the missile position. The velocity magnitude condition:

\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = v_w^2

The pointing condition (tangent toward ship at $(v_e t, H)$ ):

\frac{dy}{dx} = \frac{H - y}{v_e t - x}

Differentiating the pointing condition with respect to $t$ and using $\lambda = v_e/v_w$ , we get:

\frac{d^2x}{dy^2} \cdot \frac{H-y}{\sqrt{1 + (dx/dy)^2}} = \lambda

Setting $p = dx/dy$ , this becomes a separable first-order equation. After solving with initial conditions $x(0) = 0, x'(0) = 0$ :

x = \frac{1}{2}\left(\frac{(H-y)^{1+\lambda}}{H^\lambda(1+\lambda)} - \frac{H^\lambda(H-y)^{1-\lambda}}{1-\lambda}\right) + \frac{\lambda H}{1-\lambda^2}

When $y = H$ , the missile intercepts at $x = L = \frac{\lambda H}{1-\lambda^2}$ , at time $T = L/v_e$ .

Practice Quiz

Higher-Order Equations Quiz

Questions

Correct

Accuracy

For the equation

F(t, x^{(k)}, \ldots, x^{(n)}) = 0

where

k \geq 1

, what substitution reduces the order?

Easy

Not attempted

An autonomous equation is one where:

Easy

Not attempted

What is a first integral of a differential equation?

Easy

Not attempted

For the autonomous equation

F(x, x', x'') = 0

, what substitution is commonly used?

Medium

Not attempted

How many independent first integrals does an

n

-th order ODE have in a neighborhood of a regular point?

Medium

Not attempted

The equation

mx'' = f(x)

(where

m

is mass and

f

is force) can be integrated once to get:

Medium

Not attempted

A second-order ODE

x'' = f(t, x, x')

can be converted to a first-order system of dimension:

Medium

Not attempted

In the missile tracking problem, if the missile always points toward the target, the resulting equation is:

Medium

Not attempted

For the

n

-th order equation

x^{(n)} = f(t, x, \ldots, x^{(n-1)})

, setting

x_k = x^{(k-1)}

for

k = 1, \ldots, n

gives a system where:

Hard

Not attempted

\Phi_1, \ldots, \Phi_n

are

n

independent first integrals of the system

\frac{dx}{dt} = f(t,x)

, then the Jacobian

\frac{\partial(\Phi_1, \ldots, \Phi_n)}{\partial(x_1, \ldots, x_n)}

is:

Hard

Not attempted

Frequently Asked Questions

When should I use reduction of order?

Use reduction of order when the equation is missing certain variables: (1) If

x, x', \ldots, x^,{(k-1)}

are missing, substitute

y = x^,{(k)}

. (2) If

t

is missing (autonomous equation), substitute

y = x'

and express

x''

y(dy/dx)

. This reduces the order by one.

What is the physical meaning of a first integral?

A first integral represents a conserved quantity. For mechanical systems, the most common first integral is energy:

E = \frac{1}{2}m(x')^2 + V(x) = \text{constant}

. Other examples include angular momentum in central force problems and the Jacobi integral in rotating systems.

Why convert higher-order equations to systems?

Converting to first-order systems: (1) Provides a unified theoretical framework, (2) Makes numerical solution straightforward (most algorithms are for first-order systems), (3) Enables phase plane analysis and geometric interpretation, (4) Connects to linear algebra methods when the system is linear.

How do I find first integrals in general?

There's no universal method, but common approaches include: (1) Identify symmetries (Noether's theorem), (2) Multiply the equation by appropriate factors and integrate, (3) For Hamiltonian systems, use the Hamiltonian itself, (4) For systems with known structure (e.g., rotation), exploit the geometry.

What is the relationship between order and dimension?

n

-th order ODE is equivalent to a first-order system of dimension

n

. The "state" of the system requires

n

initial conditions:

x(t_0), x'(t_0), \ldots, x^,{(n-1)},(t_0)

. These become the initial values of the

n

-dimensional state vector.

Can all higher-order equations be reduced?

Not all equations admit useful reductions. Reduction of order works when specific variables are missing. For general equations

F(t, x, x', \ldots, x^{(n)}) = 0

, conversion to a first-order system is always possible, but this doesn't reduce complexity—it just changes the form.