Proof of Abel's Formula

Abel's formula relates the Wronskian of two solutions to the coefficient function in a second-order linear ODE. This elegant result has profound implications for the structure of solution spaces.

ProofAbel's Formula

Statement: If $y_1$ and $y_2$ are solutions of the linear differential equation:

$y'' + p(x)y' + q(x)y = 0$

then their Wronskian $W(y_1, y_2)$ satisfies:

$W(x) = Ce^{-\int p(x)dx}$

where $C$ is a constant determined by initial conditions.

Proof:

Recall the Wronskian definition:

$W(x) = y_1(x)y_2'(x) - y_1'(x)y_2(x)$

Step 1: Compute $W'(x)$ .

Using the product rule:

$W'(x) = \frac{d}{dx}[y_1y_2' - y_1'y_2]$

$= (y_1'y_2' + y_1y_2'') - (y_1''y_2 + y_1'y_2')$

$= y_1y_2'' - y_1''y_2$

The $y_1'y_2'$ terms cancel, leaving only second derivatives.

Step 2: Use the differential equation.

Since both $y_1$ and $y_2$ satisfy $y'' + p(x)y' + q(x)y = 0$ , we can solve for the second derivatives:

$y_1'' = -p(x)y_1' - q(x)y_1$

$y_2'' = -p(x)y_2' - q(x)y_2$

Step 3: Substitute into $W'$ .

$W' = y_1[-p(x)y_2' - q(x)y_2] - [-p(x)y_1' - q(x)y_1]y_2$

$= -p(x)y_1y_2' - q(x)y_1y_2 + p(x)y_1'y_2 + q(x)y_1y_2$

$= -p(x)y_1y_2' + p(x)y_1'y_2$

$= -p(x)[y_1y_2' - y_1'y_2]$

$= -p(x)W(x)$

Step 4: Solve the first-order ODE for $W$ .

We have shown that $W$ satisfies:

$\frac{dW}{dx} = -p(x)W$

This is a separable first-order linear ODE. Separating variables:

$\frac{dW}{W} = -p(x)dx$

Integrating both sides:

$\ln|W| = -\int p(x)dx + K$

where $K$ is a constant of integration. Exponentiating:

$|W| = e^K \cdot e^{-\int p(x)dx}$

Let $C = \pm e^K$ , giving:

$W(x) = Ce^{-\int p(x)dx}$

The constant $C$ is determined by evaluating $W$ at any point $x_0$ :

$C = W(x_0)e^{\int_{x_0}^{x_0} p(s)ds} = W(x_0)$

Therefore:

$W(x) = W(x_0)e^{-\int_{x_0}^x p(s)ds}$

This completes the proof. ∎

■

Remark

Key Consequences of Abel's Formula:

Sign Preservation: Since $e^{-\int p(x)dx} > 0$ always, the Wronskian never changes sign. If $W(x_0) > 0$ , then $W(x) > 0$ for all $x$ .
Zero or Never Zero: $W(x) = 0$ for all $x$ if and only if $C = 0$ (which occurs when $y_1$ and $y_2$ are linearly dependent). Otherwise, $W(x) \neq 0$ for all $x$ .
Independence from $q(x)$ : The Wronskian depends only on $p(x)$ , not on $q(x)$ . This surprising fact simplifies many calculations.

ExampleVerifying Abel's Formula

Consider $y'' - 4y' + 4y = 0$ with solutions $y_1 = e^{2x}$ and $y_2 = xe^{2x}$ .

Here $p(x) = -4$ , so Abel's formula predicts:

$W(x) = Ce^{-\int (-4)dx} = Ce^{4x}$

Direct computation:

$W = e^{2x} \cdot (e^{2x} + 2xe^{2x}) - 2e^{2x} \cdot xe^{2x}$

$= e^{4x} + 2xe^{4x} - 2xe^{4x} = e^{4x}$

Thus $C = 1$ , and the formula is verified.

ExampleApplication to Existence

If we know one solution $y_1 \neq 0$ of $y'' + p(x)y' + q(x)y = 0$ , Abel's formula helps find a second linearly independent solution.

For linear independence, we need $W \neq 0$ . Setting:

$W = y_1y_2' - y_1'y_2 = Ce^{-\int p(x)dx}$

This is a first-order linear ODE for $y_2$ in terms of the known $y_1$ . Solving it (via integrating factor or separation) yields:

$y_2 = y_1\int \frac{e^{-\int p(x)dx}}{y_1^2}dx$

This is the reduction of order formula.

Abel's formula elegantly connects the Wronskian's behavior to the differential equation's structure, providing both theoretical insight and practical computational tools.