Laplace Transforms/Change of Variable

Ordinary Differential Equatioins

L{f(t)}

integral (0, infinity): f(t)e^(-st)dt

L{e^at}

1/(s-a)

L{1}

1/s

L{dy/dt}

SL{y} - y(0)

Heaviside Function

0, t<a; 1, t>= a

L{u_a(t)}

e^-as/s

L{u_a(t)f(t-a)}

e^-asL{f(t)}

L{y’’}

s^2L{y} - sy(0) - y’(0)

L{sinwt}

w/(s²+w²)

L{coswt}

s/(s^2+w^2)

L{e^atf(t)}

F(s-a)

L{e^atsinwt}

w/((s-a)²+w²)

L{e^atcoswt}

(s-a)/((s-a)^2+w^2)

L{tf(t)}

-Df/ds

L{tsinwt}

2ws/(s²+w²)²

L{tcoswt}

(s^2-w^2)/(s^2+w^2)^2

d_a(t)

infinity, t = a; 0, else

L{d_a(t)}

e^-as

t²y’’ + aty’ + by

p’’ + (a-1)p’ + bp, s = lnt

r(t)y + a(t)y^n

z = y^(1-n), solve dz/dt, convert to y