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Recipe

Find W(x) with Abel's formula

Use when: you know a linear homogeneous equation's coefficient $p(x)$ and one value $W(x_0)$ , but not $y_1,y_2$ themselves

Goal: an explicit formula for $W(x)$ on the whole interval, without ever finding $y_1,y_2$

Taught in Abel's formula

Quick recipe

Put the equation in standard form and read off p(x)
Integrate p(x) from a fixed point $x_0$ to $x$
Exponentiate with a negative sign
Multiply by the known value W(x0)

1
Divide every term by the coefficient of $y''$ so the equation reads $y''+p(x)y'+q(x)y=0$ , then read $p(x)$ straight off the coefficient of $y'$ .

Example

For $2y''+4y'+6y=0$ , divide every term by $2$ : $y''+2y'+3y=0$ , so $p(x)=2$ .

Why: Abel's formula is stated for the standard form with a leading coefficient of exactly $1$ — reading $p$ off the un-normalized equation would scale it wrong.
2
Compute $\int_{x_0}^{x} p(t)\,dt$ ; how much work this takes depends on whether $p$ is constant or genuinely varies.

Example

With $p(x)=2$ constant and $x_0=0$ : $\int_0^x 2\,dt=2x$ .
- $p(x)$ is a constant — the integral is just that constant times $(x-x_0)$ .
- $p(x)$ genuinely varies with $x$ — integrate it as an ordinary function of $x$ .
Why: this integral is exactly the exponent Abel's formula needs — it's the total decay $p$ contributes between $x_0$ and $x$ .
3
Form $e^{-\int_{x_0}^x p(t)\,dt}$ .

Example

$e^{-2x}$ .

Why: Abel's formula derives $W$ as the solution of the decay equation $W'=-pW$ , and the minus sign belongs to that equation, not to $p$ itself — dropping it flips growth and decay.
4
Substitute the given $W(x_0)$ to get the explicit formula for $W(x)$ .

Example

With $W(0)=4$ : $W(x)=4e^{-2x}$ .

Why: $x_0$ is exactly where the formula's constant of integration gets pinned down by the one piece of data the problem actually hands you.

Common mistakes

Reading $p(x)$ off an equation before dividing to standard form (e.g. reading $p=4$ straight from $2y''+4y'+6y=0$ instead of the correct $p=2$ ).
Dropping the minus sign in the exponent.
Using a lower limit $x_0$ that doesn't match the point where $W(x_0)$ is actually known.
Trying to find $y_1,y_2$ explicitly first — Abel's formula is precisely the tool that makes that unnecessary.