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Differential Equations formula sheet

14 formulas · what each one means · a free lesson for each

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steplyclass.com/formula-sheets/differential-equations/ · free, step-by-step lessons for every formula

First-order equations

Separable
dydx = g(x)h(y) → ∫dyh(y) = ∫g(x) dx
Get all y on one side, all x on the other.Lesson: Separable equations →
Growth and decay
dydt = ky → y = y0 ekt
Lesson: Growth and decay →
Newton's law of cooling
T = Ts + (T0 − Ts)ekt
T_s = room temperature.Lesson: Newton's law of cooling →
Integrating factor
y′ + P(x)y = Q(x), μ = e∫P(x) dx
Multiply by μ; the left side is (μy)′.Lesson: The integrating factor →
Euler's method
yn+1 = yn + h f(xn, yn)
h = step size.Lesson: Euler's method →

Second-order equations

Characteristic equation
ay″ + by′ + cy = 0 → ar2 + br + c = 0
Lesson: Second order →
Two real roots
y = C1 er1 x + C2 er2 x
Lesson: Second order →
A repeated root
y = (C1 + C2 x)erx
Lesson: Second order →
Complex roots α ± βi
y = eαx(C1 cos βx + C2 sin βx)
Lesson: Second order →
Undetermined coefficients
y = yc + yp
Guess y_p in the same form as the right side.Lesson: Undetermined coefficients →

Laplace transforms

Definition
ℒ{f(t)} = ∫∞0 e−st f(t) dt
Lesson: The Laplace transform →
Table
ℒ{1} = 1s ℒ{tn} = n!sn+1 ℒ{eat} = 1s − a
Lesson: The Laplace transform →
Sine and cosine
ℒ{sin bt} = bs2 + b2 ℒ{cos bt} = ss2 + b2
Lesson: The Laplace transform →
Derivatives
ℒ{y′} = sY − y(0) ℒ{y″} = s2 Y − sy(0) − y′(0)
Lesson: Solving with Laplace →

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