Linear Differential Equations

A linear differential equation is an equation where the unknown function $ y = y(x) $ and its derivatives appear in a linear combination. Its general form is: $$ y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y = g(x) $$

A Quick Overview
Solutions to a Linear Differential Equation
Key Properties of Linear Differential Equations

A Quick Overview

The function $ y $ is the unknown in the equation and must be $ k $-times differentiable on an interval $[a, b] \subseteq \mathbb{R}$.

$$ y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y = g(x) $$

The terms $ a_i $, for $ i = 0, \dots, k-1 $, are called the coefficients of the differential equation.

These coefficients can either be constants or functions that depend on the variable $ x $.

Variable Coefficients
These are continuous functions defined on the interval $[a, b] \subseteq \mathbb{R}$. $$ y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y = g(x) $$
Constant Coefficients
These are fixed real numbers: $$ y^{(k)} + a_{k-1}y^{(k-1)} + \dots + a_1y' + a_0y = g(x) $$

The function $ g(x) $ is called the nonhomogeneous term (or forcing function).

It may be a continuous function on $[a, b]$, or simply a constant: $ g(x) = k $.

$$ y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y = k $$

When the nonhomogeneous term is zero - i.e., $ g(x) = 0 $ - the equation is called homogeneous:

$$ y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y = 0 $$

Note. The coefficient of the highest-order derivative is typically normalized to 1. This is always possible because the leading coefficient $ a_k $ must be nonzero: $$ a_k y^{(k)} + a_{k-1}y^{(k-1)} + \dots + a_1y' + a_0y = g(x) $$ By dividing both sides by $ a_k $, we obtain: $$ \frac{a_k}{a_k} y^{(k)} + \frac{a_{k-1}}{a_k} y^{(k-1)} + \dots + \frac{a_1}{a_k} y' + \frac{a_0}{a_k} y = \frac{g(x)}{a_k} $$ Which simplifies to: $$ y^{(k)} + \frac{a_{k-1}}{a_k} y^{(k-1)} + \dots + \frac{a_1}{a_k} y' + \frac{a_0}{a_k} y = \frac{g(x)}{a_k} $$ This is the normal form of the differential equation: $$ y^{(k)} = \frac{g(x)}{a_k} - \frac{a_{k-1}}{a_k} y^{(k-1)} - \dots - \frac{a_1}{a_k} y' - \frac{a_0}{a_k} y $$

Why is it called "linear"?

A differential equation is called linear because the operator $ L $, which maps the function $ y $ to the left-hand side of the equation, is a linear operator:

$$ L(y) = y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y $$

Note. The expression on the left is a linear combination of the function and its derivatives: $$ y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y $$ This defines a linear operator $ L $ that maps functions from the space $ C^k(a, b) $ (functions differentiable $ k $ times) into $ C^0(a, b) $ (continuous functions): $$ L : C^k(a, b) \rightarrow C^0(a, b) $$ Being linear, it satisfies the properties: $$ L(y + u) = L(y) + L(u), \quad \forall y, u \in C^k $$ $$ L(\lambda y) = \lambda L(y), \quad \forall \lambda \in \mathbb{R}, \ y \in C^k $$

In general, for any functions $ y $, $ u $ and scalars $ \alpha $, $ \beta $, the linearity property ensures: $$ L(\alpha y + \beta u) = \alpha L(y) + \beta L(u) $$

This structure is fundamental when solving linear differential equations, as it allows the construction of new solutions through linear combinations.

Solutions to a Linear Differential Equation

A solution to a linear differential equation is a function $ y = y(x) $, differentiable up to order $ k $ on an interval $[a, b]$, that satisfies: $$ y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y = g(x) $$

This solution is often referred to as the integral of the equation.

Note. A linear differential equation may admit multiple solutions. The complete set of all such solutions is called the general solution (or general integral).

Key Properties of Linear Differential Equations

Some important properties of linear differential equations include:

If $ y(x) $ and $ u(x) $ are solutions of the same homogeneous linear equation, then any linear combination is also a solution: $$ L(y) = L(u) = 0 \Rightarrow L(\alpha y + \beta u) = \alpha L(y) + \beta L(u) = 0 $$
If $ y(x) $ and $ u(x) $ both solve the same nonhomogeneous equation, then any linear combination is a solution of the same equation only under special conditions: $$ L(y) = L(u) = g \Rightarrow L(\alpha y + \beta u) = \alpha g + \beta g = (\alpha + \beta)g \neq g \ \text{in general} $$ However, their difference always solves the associated homogeneous equation: $$ L(y - u) = L(y) - L(u) = g - g = 0 $$
Note. This implies that every solution of a nonhomogeneous equation $$ y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y = g(x) $$ can be written as the sum of a particular solution of the nonhomogeneous equation and the general solution of the corresponding homogeneous equation: $$ y^{(k)} + a_{k-1}(x)y^{(k-1)} + \dots + a_1(x)y' + a_0(x)y = 0 $$
If all the coefficients are defined and continuous on the interval $ (a, b) $, then the solutions are also defined throughout $ (a, b) $.

And so on.