finding another particular solution of linear ODE
If one knows one particular solution (http://planetmath.org/SolutionsOfOrdinaryDifferentialEquation) of (1), it’s possible to derive from it via two quadratures another solution , linearly independent on ; thus one can write the general solution
of that homogeneous differential equation.
We will now show the derivation procedure.
which renders (1) to
Here one can choose , whence the first addend vanishes, and (3) gets the form
This equation may be written as , which is integrated to
A new integration results from this the general solution of (4):
Thus by (2), we have obtained the wanted other solution
which is clearly linearly independent on y_1(x).
Consequently, we can express the general solution of the differential equation (1) as
where and are arbitrary constants.
- 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
|Title||finding another particular solution of linear ODE|
|Date of creation||2014-02-28 14:31:42|
|Last modified on||2014-02-28 14:31:42|
|Last modified by||pahio (2872)|