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Differential equations — Formula Sheet

4 formulas from Differential Equations, each with what it means, when to use it and the trap it sets.

Separable equation

\frac{d y}{d x} = g (x) h (y) \Rightarrow \int \frac{d y}{h ( y )} = \int g (x) d x

What it says: Get all the y on one side and all the x on the other, then integrate both.
When to reach for it: The right-hand side factors cleanly into an x part times a y part.
Watch out: One constant of integration is enough for both sides, and dividing by h(y) can lose the constant solution h(y) = 0.

y^{'} = x y \Rightarrow ln ∣ y ∣ = \frac{x ^{2}}{2} + C

Exponentiate at the end to get y = Ae^{x²/2}.

Integrating factor

y^{'} + P (x) y = Q (x), μ = e^{\int P d x}, (μ y)^{'} = μ Q

What it says: Multiply through by μ and the left side collapses into a single product derivative.
When to reach for it: First-order linear equations that will not separate.
Watch out: The equation must be in standard form with the coefficient of y′ equal to 1 before you read off P.

y^{'} + 2 y = e^{x} : μ = e^{2 x} \Rightarrow y = \frac{1}{3} e^{x} + C e^{- 2 x}

The whole method exists to make the left side an exact derivative.

Characteristic equation

a y^{''} + b y^{'} + cy = 0 \Rightarrow a r^{2} + b r + c = 0

What it says: Guess y = e^{rx} and the differential equation becomes an ordinary quadratic.
When to reach for it: Second-order linear equations with constant coefficients.
Watch out: Three cases: distinct real roots give two exponentials, a repeated root needs an extra factor of x, and complex roots a ± bi give e^{ax}(C₁cos bx + C₂sin bx).

y^{''} - 3 y^{'} + 2 y = 0 \Rightarrow r = 1, 2 \Rightarrow y = C_{1} e^{x} + C_{2} e^{2 x}

Two distinct real roots, so two independent exponentials.

Laplace transform of a derivative

L {f^{'}} = s F (s) - f (0)

What it says: Differentiation becomes multiplication by s, with the initial condition folded in.
When to reach for it: Initial-value problems, especially with discontinuous or impulsive forcing.
Watch out: The initial values are part of the transform. Dropping f(0) silently solves a different problem.

L {f^{''}} = s^{2} F (s) - s f (0) - f^{'} (0)

Each extra derivative adds one more initial-condition term.

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