Laplace Transforms
The definition
The Laplace transform turns a function of time f(t) into a function of a complex variable s, trading a differential equation in t for an algebraic one in s:
L{f(t)}=F(s)=∫0∞e−stf(t)dtStandard transforms
L{1}=s1,L{tn}=sn+1n!,L{eat}=s−a1L{sinat}=s2+a2a,L{cosat}=s2+a2sThe transform is linear: the transform of a sum is the sum of the transforms, and constants pull straight out, which is what makes it usable on multi-term expressions at all.
The first shifting theorem
Multiplying a function by eat shifts its transform by a:
L{eatf(t)}=F(s−a)Worked example: find L{e3tsin4t}.
Start from L{sin4t}=s2+164, then replace s with s−3:
L{e3tsin4t}=(s−3)2+164Worked example: solving a linear ODE
Solve dtdy+2y=e−t given y(0)=0.
Solution: Taking the Laplace transform of both sides (using L{y′}=sY(s)−y(0)) turns the ODE into an algebraic equation in Y(s):
sY(s)−0+2Y(s)=s+11⇒(s+2)Y(s)=s+11Y(s)=(s+1)(s+2)1Splitting into partial fractions, (s+1)(s+2)1=s+1A+s+2B. Setting s=−1 gives A=1; setting s=−2 gives B=−1:
Y(s)=s+11−s+21Each term is already a standard transform, so reading the inverse straight off the table:
y(t)=e−t−e−2tChecking: y(0)=1−1=0 as required, and y′+2y=(−e−t+2e−2t)+(2e−t−2e−2t)=e−t, matching the original equation.
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Laplace transforms and Fourier series are the two core tools engineering math uses to turn a hard problem in the time domain into an easier one somewhere else.