Laplace Transforms and other BS rules of differential equations

Laplace transform of f(t) = 1

1/s

Laplace transform of f(t) = t

1/(s²)

Laplace transform of f(t) = t^n

n!/(s^(n+1)); n > 0

Laplace transform of f(t) = sin(kt)

k/(s²+k²)

Laplace transform of f(t) = cos(kt)

s/(s²+k²)

Laplace transform of f(t) = e^(at)

1/(s-a)

Laplace transform of f(t) = sinh(kt)

k/(s²-k²)

Laplace transform of f(t) = cosh(kt)

s/(s²-k²)

Laplace transform of y’(t)

sY(s) - y(0)

Laplace transform of y’’(t)

s²Y(s) - sy(0) - y’(0)

Laplace transform of y^(n)(t)

s^nY(s) - s^(n-1)y(0)-….- y^(n-1)(0); n > 0

Laplace transform of f(t) = e^(at)*f(t)

F(s-a)

Laplace transform of f(t) = u(t-a)

(e^-(as))/s

Laplace transform of f(t) = f(t-a)u(t-a)

(e^-(as))F(s)

Laplace transform of f(t) = g(t)u(t-a)

e^-(as)(L{g(t+a)}

Laplace transform of f(t) = t^(n)(f(t))

(-1)^n(d^n/ds^n)F(s)

Laplace transform of f(t) = f*g = integral from 0 to t of f(tau) times g(t-tau) d(Tau)

F(s)G(s)

Laplace transform of f(t) = integral of f(tau) d(tau) from 0 to t

F(s)/s

Laplace transform of f(t) = (e^(at))cos(bt)

(s-a)/((s²-a²)+b²)

Laplace transform of f(t)= (e^(at)sin(bt))

b/((s²-a²)+b²)

Laplace transform of f(t) = (t^n)(e^at)

n!/(s-a)^n+1

Laplace transform of f(t) = f(ct)

(1/c)L{f(s/c)}

Laplace transform of f(t) = f^n(t)

(s^n)L{f(t)}-s^(n-1)f(0)-…-sf^(n-2)(0)-f^(n-1)(0)

Partial fraction form given this term in the denominator: (s+a)

A/(s+a)

Partial fraction form given this term in the denominator: (s+a)²

(A/(s+a))+(B/(s+a)²)

Partial fraction form given this term in the denominator: (s+a)³

(A/(s+a)) + (B/(s+a)²) + (C/(s+a)³)

Partial fraction form given this term in the denominator: (s² ± k²)

(As + B) / (s² ± k²)

Partial fraction form given this term in the denominator: (s² ± k²)²

((As + B) / (s² ± k²)) + ((Cs + D) / (s² ± k²))

Partial fraction form given this term in the denominator: (s² + bs+c) irreducible

(As + B) / (s² + bs + c)

Partial fraction form given this term in the denominator: (s²+bs+c)²

((As + B) / (s²+bs+c)) + ((Cs + D) / (s²+bs+c)²)

How to complete the square when given irreducible quadratics

Special form: (s ± a)²±k², halve b, square it, to create the new quadratic, and subtract it from c.
Ex: s² + bs + c = (s + (b/2))² + c - (b/2)²