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