Proof of Abel's Formula for Higher-Order Equations

Abel's formula extends to $n$ -th order linear ODEs, expressing the Wronskian of any set of $n$ solutions in closed form.

Statement

Theorem3.3Abel's formula (general)

For the $n$ -th order linear ODE $y^{(n)} + p_{n-1}(t)y^{(n-1)} + \cdots + p_0(t)y = 0$ on an interval $I$ , if $y_1, \ldots, y_n$ are any $n$ solutions, their Wronskian satisfies

$W(y_1, \ldots, y_n)(t) = W(t_0)\exp\left(-\int_{t_0}^t p_{n-1}(s)\,ds\right)$

for any $t_0 \in I$ .

Proof

Step 1: Differentiate the Wronskian.

The Wronskian is $W = \det\begin{pmatrix} y_1 & \cdots & y_n \\ y_1' & \cdots & y_n' \\ \vdots & & \vdots \\ y_1^{(n-1)} & \cdots & y_n^{(n-1)}\end{pmatrix}$ .

Differentiating a determinant with respect to $t$ gives a sum of $n$ determinants, each obtained by differentiating one row. When we differentiate any row $k < n$ (the row of $y_j^{(k)}$ becomes $y_j^{(k+1)}$ ), the resulting matrix has two identical adjacent rows, so its determinant is $0$ .

The only nonzero contribution comes from differentiating the last row ( $k = n-1$ ):

$W' = \det\begin{pmatrix} y_1 & \cdots & y_n \\ y_1' & \cdots & y_n' \\ \vdots & & \vdots \\ y_1^{(n)} & \cdots & y_n^{(n)}\end{pmatrix}.$

Step 2: Use the ODE to replace $y_j^{(n)}$ .

Since each $y_j$ satisfies the ODE: $y_j^{(n)} = -p_{n-1}y_j^{(n-1)} - p_{n-2}y_j^{(n-2)} - \cdots - p_0 y_j$ .

Substituting into the last row:

$W' = \det\begin{pmatrix} y_1 & \cdots & y_n \\ \vdots & & \vdots \\ y_1^{(n-2)} & \cdots & y_n^{(n-2)} \\ -p_{n-1}y_1^{(n-1)} - \cdots - p_0 y_1 & \cdots & -p_{n-1}y_n^{(n-1)} - \cdots - p_0 y_n\end{pmatrix}.$

The last row is $-p_{n-1}\cdot(\text{row } n) - p_{n-2}\cdot(\text{row } n-1) - \cdots - p_0\cdot(\text{row } 1)$ of the original Wronskian matrix. By properties of determinants (adding multiples of other rows to a row does not change the determinant), only the $-p_{n-1}$ term contributes:

$W' = -p_{n-1}(t) W.$

Step 3: Solve the first-order ODE for $W$ .

The equation $W' = -p_{n-1}(t)W$ is separable with solution

$W(t) = W(t_0)\exp\left(-\int_{t_0}^t p_{n-1}(s)\,ds\right). \qquad \blacksquare$

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Applications

ExampleWronskian without computing solutions

For $y'' - 2ty' + 2y = 0$ (Hermite's equation with $n=1$ ), $p_1(t) = -2t$ so

$W(t) = W(0)\exp\left(\int_0^t 2s\,ds\right) = W(0)e^{t^2}.$

Without solving the ODE, we know the Wronskian of any two independent solutions grows like $e^{t^2}$ .

ExampleReduction of order via Abel's formula

If $y_1$ is a known solution of $y'' + p(t)y' + q(t)y = 0$ , Abel's formula gives $W(y_1, y_2) = Ce^{-\int p\,dt}$ . Since $W = y_1 y_2' - y_1'y_2$ , this is a first-order ODE for $y_2$ :

$y_1 y_2' - y_1'y_2 = Ce^{-\int p\,dt} \implies \left(\frac{y_2}{y_1}\right)' = \frac{C}{y_1^2}e^{-\int p\,dt}.$

Integrating: $y_2(t) = y_1(t)\int \frac{e^{-\int p\,ds}}{y_1(s)^2}\,ds$ .

RemarkLiouville's formula

Abel's formula is also known as Liouville's formula. It generalizes to systems: for $\mathbf{x}' = A(t)\mathbf{x}$ , the determinant of the fundamental matrix satisfies $W'(t) = \text{tr}(A(t))W(t)$ , giving $W(t) = W(t_0)\exp(\int_{t_0}^t \text{tr}(A)\,ds)$ .