image equation
In solving an initial value problem leading to an ordinary differential
equation, the Laplace transform offers often a way to simplify the equation:
both sides are Laplace transformed. The transformed equation, the so-called
image equation, is in many cases simplier than the original differential
equation, since it does not contain the derivatives of the unknown function
. From the image equation one may solve the Laplace transform of
and then inverse transform getting .
Let’s consider e.g. the ordinary ’th order linear differential equation
(1) |
subject to the initial conditions
(2) |
Due to the linearity of the Laplace transform the image equation of (1) is
(3) |
Denote and . We put into (3) the expressions of the Laplace transforms of the derivatives on the left hand side (see “Laplace transforms of derivatives (http://planetmath.org/LaplaceTransformsOfDerivatives)”) getting
This equation is simplified to
For brevity, denote in the last equation the polynomial multiplier of by and the sum preceding by . Then the equation can be written as
i.e.
(4) |
The function defined by (4) is the Laplace transform of
the solution of the differential equation (1) which
satisfies the initial conditions (2). If we now find a function
the Laplace transform of which is the function
defined by (4), then will do for due to the
uniqueness property of Laplace transform expressed in the entry
“Mellin’s inverse formula (http://planetmath.org/MellinsInverseFormula)”.
If we seek the solution of (1) satisfying the zero initial
conditions
then and
i.e.
Example. The 4’th order differential equation
(5) |
should be solved with the initial conditions
The image equation of (5) is
i.e.
Thus one needs to determine the inverse Laplace transform of
(6) |
The zeroes of the numerator are the eighth roots of unity , , , , in other words the complex numbers . By the special case (3) of the Heaviside formula, the first addend of (6) corresponds the original function
Utilizing also the general Heaviside formula (http://planetmath.org/HeavisideFormula) (1), one can get from (6) the result
References
- 1 N. Piskunov: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele. Teine köide. Viies trükk. Kirjastus Valgus, Tallinn (1966).
Title | image equation |
---|---|
Canonical name | ImageEquation |
Date of creation | 2014-03-20 20:16:56 |
Last modified on | 2014-03-20 20:16:56 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 13 |
Author | pahio (2872) |
Entry type | Topic |
Classification | msc 34A05 |
Classification | msc 44A10 |