Linear Differential Equations

What This Chapter Covers

Section	What you will learn
3.1 Linear Differential Operators	Writing $L[y]=f(x)$ with the operator $L$, and the linearity property that drives everything else.
3.2 Linear Independence and the Wronskian	What it means for functions to be independent, and the determinant that tests it.
3.3 Abel's Formula and Fundamental Sets	Why the Wronskian of solutions is either always zero or never zero, and what a fundamental set is.
3.4 The General Solution and Variation of Parameters	The structure $y=y_h+y_p$, and a method that produces $y_p$ for any forcing function.

The Central Facts

1. Superposition. If $y_1$ and $y_2$ solve the homogeneous equation $L[y]=0$, so does $c_1{y}_1+c_2{y}_2$ for any constants. Solutions of a homogeneous linear equation form a vector space.
2. Dimension. That vector space has dimension exactly $n$ for an equation of order $n$. So $n$ linearly independent solutions capture every solution there is.
3. The Wronskian test. For solutions of the same linear equation, independence can be checked at a single point: the Wronskian is either identically zero or never zero on the interval.
4. Decomposition. The general solution of $L[y]=f$ is $y=y_h+y_p$, where $y_h$ is the general solution of $L[y]=0$ and $y_p$ is any one solution of the full equation.

The practical consequence is a two-part strategy that recurs in every later chapter: find $n$ independent solutions of the homogeneous equation, then find one particular solution of the full equation. The chapters that follow are largely about carrying out those two steps for specific classes of equations.

Why Linearity Matters

Nonlinear equations have no comparable theory. Their solutions do not add, there is no notion of dimension, and singular solutions can appear from nowhere. For linear equations, by contrast, the language of linear algebra applies almost word for word:

• The operator $L$ plays the role of a linear transformation.
• Solutions of $L[y]=0$ form the kernel of that transformation.
• A fundamental set of solutions is a basis for the kernel.
• The Wronskian plays the role of a determinant testing independence.
• The decomposition $y=y_h+y_p$ is the standard description of the solution set of a nonhomogeneous linear system.

If you have met these ideas in a linear algebra course, this chapter will feel familiar. If not, it works equally well as a first encounter with them, in a setting where the "vectors" are functions.

Frequently Asked Questions

Why does an $n$th order equation need exactly $n$ independent solutions?

Because the solution space of $L[y]=0$ has dimension $n$. Intuitively, specifying a solution requires specifying y(x_0), y'(x_0), \dots, y^{(n-1)}(x_0), which is $n$ pieces of data, and the existence and uniqueness theorem says every such choice determines exactly one solution. So the space of solutions is in one-to-one correspondence with ${ℝ}^n$.

Does this chapter apply to equations with variable coefficients?

Yes, entirely. Nothing here assumes constant coefficients. The theory holds whenever the coefficients $P_1(x),\dots ,P_n(x)$ are continuous on the interval. What constant coefficients buy you is a method for actually finding the $n$ independent solutions, which is the subject of Chapter 4.

Is the Wronskian test valid for arbitrary functions?

Only in one direction. A nonzero Wronskian always implies independence, for any functions. The converse holds only for solutions of a common linear differential equation. There are independent functions whose Wronskian vanishes identically, so the test must be applied with care outside the setting of this chapter.