Laplace transform
y'' and y'
-y'(0) - sy(0) + s^2Y(s) -y(0) + sY(s)
from f(x) world to laplace world
1. write f(x) in terms of unit step function by decomposing it. 2. And then translate the unit step function. =>Enter: g - world i.e. [x-2]u(x-1) <=> g(x-1)u(x-1) where f(x) = [x-2] 3. Then. apply the laplace to the part in the form of g(x-c)u(x-c) <=> G(s) * e^(-cs) 4. Figure out what G(s) by letting t = x - c => x = t + c 5. now, go from g world to f world => g(t) = g(x-c) = f(x) 6. substitute x = t + c into the g-world => g(x) = x -1 i.e. t + 1 - 2 = g(x) = t -1 => g(x) = x-1 7. now replace f(x) with g(x) and perform the laplace transform of it. L[ x - 1 ] and voila, now you have gone from f world to Laplace world.
1/ (s+a)
1/ a - e^(-cs) / a
e^(ax)
1/(s-a)
1
1/s
x*sin(Bx)
2Bs / (s² + B²)²
e^(ax)*sinh(Bx)
B / ((s-alpha)^2 - B^2)
e^(ax)*sin(Bx)
B / (s-a)² + B²
sin(Bx)
B / (s² + B²)
sinh(Bx)
B / s^2 - B^2
inverse laplace transformation
If F(s) is a given transform and if the function f, is continous on [0, oo), has the property that L[f(x)] = F(s), then f is called the ______________ of F(s) and is denoted by f(x) = L⁻¹[F(s)]
when e^(-as) is observed
Use exponentiation to find the inverse laplace transform. L⁻¹[e^(-cs)] = f(x-c)u(x-c)
c
c / s
laplace transformation
can be used to turn linear constant coefficient differential equations into algebraic equations. Definition: Let 'f' be a continuous function on the interval [0,oo). The Laplace transform of 'f', denoted by L[f(x)] = F(s) = the integral of e^(-sx)*f(x)dx from zero to infinity The domain of F is the set of all real numbers s for which the improper integral converges.
x^n * e^(ax)
n! / (s - a)^n+1
x^(n) * e^(ax)
n! / (s-a)^(n+1)
x^n
n! / sⁿ⁺¹ n = 1,2, ...
synthetic division
r = constant value
e^(ax)*cos(Bx)
s - a / (s-a)² + B²
e^(ax)*cosh(Bx)
s - alpha / ((s-alpha)^2 - B^2)
cos(Bx)
s / (s² + B²)
cosh(Bx)
s / s^2 - B^2
x*cos(Bx)
s² - B² / (s² + B²)²
When a / x is observed
use euler cauchy to solve the nth order differential equation. let y = x^(r) and take the derivative of it.
long division
when given r = ai value