PERTEMUAN XVI
Aplikasi Transformasi Laplace.
Application to Differential Equations
Consider the linear differential equation with constant coefficients
under the initial conditions
The Laplace transform directly gives the solution without going through the
general solution. The steps to follow are:
(1) Evaluate the Laplace transform of the two sides of the equation (C);
(2) Use Property 14 (see Table of Laplace Transforms)
;
(3) After algebraic manipulation, write down
;
(4) Make use of the properties of the inverse Laplace transform
solution
y(t).
, to find the
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Jawab :
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Example: Find the solution of the IVP
,
where
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.
Solution: Let us follow these steps:
(1) Using properties of Laplace transform, we get
s2Y(s) – sf(0) – f’(0) + 3(sY(s) – f(0)) + 2Y(s) = L(g(t))
(2) We have
,
where
.
Since
,
we get
;
(3)
Inverse Laplace:
Using partial decomposition technique we get
,
which implies (see Table of Laplace Transforms)
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Since
,
which gives (see Table of Laplace Transforms)
,
and
Hence,
Example 1: Solve using Laplace Transform
Solution: First, apply the Laplace Transform
Knowing that
,
and
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we get
After easy algebraic manipulations we get
,
which implies
Next, we need to use the inverse Laplace.
We have (see the table)
For the second term we need to perform the partial decomposition
technique first.
.
We get
Hence, we have
Since (see the table)
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and (see the table)
Finally, we have
Example: Find the solution of the IVP
Solution. We follow these steps:
(1)
We apply the Laplace transform
,
where
. Hence,
;
(2)
Inverse Laplace:
Since
,
and
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we get
Example: Find the solution to
Solution: Apply the Laplace transform to get
where
. Hence,
We then rewrite Y(s) to get
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A linear ordinary differential equation ( ODE ) of order n is an equation as :
c0(x) * y(x) + c1(x) * y'(x) + ... + cn(x) * y(n)(x) = Q(x) ,
where y(n) = (dny)/(dxn) .
Examples of ODE :
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For the ODE :
You may type :
y'(x)+p(x)y(x)=q(x)
y'+p(x)*y=q(x)
( Euler )
x y''(x)+axy'(x)+by(x)=S(x)
x^2*y''+a*x*y'+b*y=S(x)
( Chebyshev )
(1-x )y''(x)-xy'(x)+a2y(x)=0
(1-x^2)*y''-x*y'+a^2y=0
y'(x)-2y(x) =
sqrt(x)*y'-2*y=exp(x) ou
(x)^(1/2)y'-2y=e^x
2
2
The derivate functions of y have to be
The unknowed function can be : y(x) , y(t) or x(t).
typed
y',y'',y''',y'''',...
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Solve each with Laplace transforms:
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1. y'( t ) - y( t ) = 0, y( 0 ) = 7.
2. y''( t ) - y'( t ) + y( t ) = et, y( 0 ) = 0, y'( 0 ) = 1.
3. y'( t ) + y( t ) = 5 Unit Step ( t - 1), y( 0 ) = 0.
TERIMA KASIH
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