Non-homogeneous — variation of parameters: Foundations

This page assumes you have seen nothing. We name every mark on the page, draw the picture it stands for, and say why the topic can't live without it. Read top to bottom; each item leans on the one above.

1. The function and its name $y (x)$

Picture. Look at figure s01. Slide a pointer along the horizontal axis (that's $x$ ). The height of the curve above that spot is the value $y$ . The whole curve is the function.

Why the topic needs it. An "ordinary differential equation" (ODE) is a puzzle whose answer is a whole function $y (x)$ , not a single number. So before anything else we must be comfortable that the unknown is a curve.

2. The prime — a derivative $y^{'}$

Picture. Look at figure s02. At any point the little tangent line touches the curve; its tilt is $y^{'}$ . Now watch that tilt as you slide right: on a curve that bends upward (a valley shape) the tangent starts tilting down, flattens, then tilts up — the tangent line rotates counter-clockwise. The speed of that rotation is $y^{''}$ . A bigger $y^{''}$ means the tangent swings faster, i.e. the curve bends more sharply.

Why the topic needs it. The whole equation $y^{''} + p y^{'} + q y = g$ is built entirely from $y, y^{'}, y^{''}$ . Without the prime we cannot even write the problem.

3. The equation $y^{''} + p y^{'} + q y = g$

"Second-order" because the highest prime is $y^{''}$ (two primes).
"Linear" because $y, y^{'}, y^{''}$ appear only to the first power — no $y^{2}$ , no $sin y$ .

Why the topic needs it. This is the very equation variation of parameters solves. The left side is the machine; the right side is what we want it to print.

4. Homogeneous vs non-homogeneous, and $g (x)$

Picture. See figure s03. Left panel: a pendulum swinging freely — homogeneous. Right panel: a hand periodically nudging it — that nudge is $g (x)$ , non-homogeneous.

Related, faster-but-narrower tool: Method of Undetermined Coefficients handles only special pushes.

5. The two building-block solutions $y_{1}, y_{2}$

Why "two"? Second-order $\Rightarrow$ two independent choices to make (like choosing a starting position and a starting speed). Getting these is the job of Second-order linear homogeneous ODE, often via Reduction of Order when you already know one.

Why the topic needs it. The whole answer is built out of $y_{1}, y_{2}$ . They are the raw lumber.

6. Constants vs functions: $c_{1}, c_{2} \to u_{1} (x), u_{2} (x)$

Picture. Figure s04. Top: two knobs $c_{1}, c_{2}$ frozen. Bottom: the same knobs, but now hands turn them as $x$ advances — the "variation of the parameters."

7. The determinant bars and the Wronskian $W$

Plain meaning. $W$ is a "fairness scale." If $W \neq = 0$ , $y_{1}$ and $y_{2}$ are truly different building blocks; if $W = 0$ , one is secretly a copy of the other and the method jams (you'd divide by zero). Full story: Wronskian.

Why the topic needs it. $W$ sits in the denominator of the final formula — it is literally the divisor when we solve for the strengths.

8. Solving two-equations-two-unknowns: Cramer's Rule

The setup produces a little system: ${u_{1}^{'} y_{1} + u_{2}^{'} y_{2} = 0 u_{1}^{'} y_{1}^{'} + u_{2}^{'} y_{2}^{'} = g$

Two unknowns here are $u_{1}^{'}$ and $u_{2}^{'}$ (the rates the strengths change).

The minus on $u_{1}^{'}$ is not a mystery — it comes from the $- b c$ in the determinant when the right side $(0, g)$ lands in the first column.

Function $y (x)$ (§1) gives us something to differentiate.
Primes $y^{'}, y^{''}$ (§2) let us write rates and curvature.
Together they form the second-order linear ODE (§3).
Splitting off the push distinguishes homogeneous vs $g$ (§4).
The homogeneous side supplies two building blocks $y_{1}, y_{2}$ (§5).
We swap constants for functions $u_{1}, u_{2}$ (§6) to reach the push.
Independence of the blocks is measured by the Wronskian $W$ (§7).
Steps 6–7 feed a 2×2 system solved by Cramer's Rule (§8), giving $u_{1}^{'}, u_{2}^{'}$ .
Integrating (§9) turns those rates into $u_{1}, u_{2}$ — the particular solution.
Standard form (§10) guards the whole pipeline by fixing $g$ first.

Equipment checklist

Test yourself — cover the right side.

What does

y^{'}

measure on a curve?

The slope (steepness) at each point;

y^{''}

is how fast that slope changes (how sharply the curve bends).

y^{''} + p y^{'} + q y = g

, which piece is "the push"?

g (x)

, the forcing on the right-hand side.

What makes an equation homogeneous?

Its right-hand side is

0

(no external push).

Why do we need exactly two solutions

y_{1}, y_{2}

A second-order equation has two independent free choices, so its free motion is a blend of two building blocks.

What is the single swap that defines variation of parameters?

Replace the constant strengths

c_{1}, c_{2}

with varying functions

u_{1} (x), u_{2} (x)

Compute the

2 \times 2

determinant

a c b d

a d - b c

How do you tell determinant bars from absolute-value bars?

Absolute value wraps a single number (

∣ - 3 ∣ = 3

, never negative); a determinant wraps a grid of four numbers (

a d - b c

, may be negative).

Write the Wronskian of

y_{1}, y_{2}

W = y_{1} y_{2}^{'} - y_{2} y_{1}^{'}

What does

W = 0

warn you about, and why does one check suffice?

The two solutions are not independent and the method breaks down; Abel's identity

W = W (x_{0}) e^{- \int p d x}

shows

W

is either zero everywhere or nowhere on the interval.

Cramer's rule gives

u_{1}^{'}

and

u_{2}^{'}

as?

u_{1}^{'} = - y_{2} g / W

and

u_{2}^{'} = + y_{1} g / W

Why integrate after Cramer's rule, and why drop the

+ C

Integrating undoes the prime to recover

u_{1}, u_{2}

; the constants would only add

C_{1} y_{1} + C_{2} y_{2}

, a homogeneous piece already counted in

c_{1} y_{1} + c_{2} y_{2}

Write the final formula for

y_{p}

y_{p} = - y_{1} \int (y_{2} g / W) d x + y_{2} \int (y_{1} g / W) d x

What must you check before reading off

g

Put the ODE in standard form (leading coefficient

1

) by dividing through.

Connections

Wronskian — the fairness scale $W$ built in Section 7.
Cramer's Rule — the shortcut that solves the two-equation system in Section 8.
Second-order linear homogeneous ODE — where the building blocks $y_{1}, y_{2}$ come from.
Reduction of Order — a way to find $y_{2}$ when only $y_{1}$ is known.
Method of Undetermined Coefficients — the faster, narrower cousin for special $g (x)$ .
Green's function — recasts this same idea as an integral kernel.

Foundations — Non-homogeneous — variation of parameters