TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Existence and uniqueness for y''=-y with prescribed initial data

Statement

For $a,b\in\mathbb R$ , the function $y(x)=a\cos x+b\sin x$ is the unique twice differentiable function on $\mathbb R$ satisfying $y''=-y,\qquad y(0)=a,\qquad y'(0)=b.$

Facts & Assumptions

Given: Reals $a,b$ and a twice differentiable solution $z$ of the displayed initial-value problem.

[L1]

$(\sin x)'=\cos x$ , $(\cos x)'=-\sin x$ , $\sin0=0$ , and $\cos0=1$ (The derivatives of sine and cosine are cosine and minus sine).

[L2]

The algebra of derivatives differentiates sums and products (Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$ , $(\alpha f)'(c) = \alpha f'(c)$ , $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$ , and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$ ).

[L3]

A differentiable function with zero derivative on $\mathbb R$ is constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant).

Proof

technique · direct

1.1

By [L1], $y=a\cos+b\sin$ satisfies $y''=-y$ , $y(0)=a$ , and $y'(0)=b$ .

L1algebra

2.1

Put $h=z-y$ . Then $h''=-h$ , $h(0)=h'(0)=0$ , and $E:=h^2+(h')^2$ has derivative $E'=2h(h+h'')=0$ .

step 1.1L2algebra

3.1

Thus $E$ is constantly $E(0)=0$ ; as both squares are nonnegative, $h=0$ everywhere and $z=y$ .

step 2.1L3algebra∎

Depends on

Used by

The addition formulas for sine and cosine Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 55 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)
C. Schmeiser, Introduction to Analysis (standard reference, not scraped)