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### Course: Differential equations>Unit 3

Lesson 3: Laplace transform to solve a differential equation

# Laplace transform solves an equation 2

Second part of using the Laplace Transform to solve a differential equation. Created by Sal Khan.

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• The Laplace transform is not always 1 to 1. For example |t| and t have the same laplace transform
• L{t} and L{|t|} are the same I think. So they are not 2 different functions, they're actually the same thing when we're doing an integral from 0 to infinity.
• At , we found that y = 9 ( e ^ ( -2t ) ) - 7 (e ^ - ( 3t ) ) but this is just "A" solution, not the GENERAL solution. Do the laplace transforms only give us "A" solution?
Edit: Because we were given initial values, we were finding a particular solution anyways...
• Can I use Laplace Transform to Solve Differential Equations where I don't have initial conditions ?
Or what is a method to get out or calculate the initial conditions of a system ? Think for example about the damped motion of suspensions in mechanical engineering...
• The problem lies in a fact, that a general solution to a differential equation is not a function, it is a set of functions. Initial values just narrow down the choices to one particular function the so called particular solution.
(1 vote)
• When the factors are linear, it is easier to do partial fraction decomposition by plugging the zeros into both sides.
• At ..How does Sal get the roots of the polynomial in the denominator so easily and fast without using the quadratic formula??
• I think he was thinking "what two numbers add to 5 and multiply to 6"
(1 vote)
• Despite knowing that Sal's solution is obviously correct, Sal should have still checked to see if y'', y', and y grow slower than e^(sx) as x approaches infinity since the definite integral in the transform grows to infinity otherwise. Also since s on the Laplace transform is only valid for s > 0, does that mean our solution is only valid for x > 0?
(1 vote)
• Not at all. Just because s > 0 doesn't mean that x > 0. Notice that x has in fact dropped out of the solution, since we integrated over all x. s is a frequency variable.
(1 vote)
• BUT, the Laplace transform of the "Dirac delta function" and of "s" is both "1"?
(1 vote)
• At Sal claims the Laplace Transformation is a 1-1 transformation, but surely the L{t} is the same as L{|t|}? Also the same would apply to L{t^2} and L{sign(t)t^2}. So when can this be justified and when not?
(1 vote)
• what if its homogenous with IC as y(0)=0?
(1 vote)
• can you make this video in articles .. please
(1 vote)