on the 3 order linear differential equation - Kathmandu University
on the 3 order linear differential equation - Kathmandu University
on the 3 order linear differential equation - Kathmandu University
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KATHMANDU UNIVERSITY JOURNAL OF SCIENCE, ENGINEERING AND TECHNOLOGY<br />
VOL. 8, No. II, DECEMBER, 2012, 7-10<br />
The expressi<strong>on</strong> (3) for<br />
can be c<strong>on</strong>structed via method of variati<strong>on</strong> of parameters of Euler<br />
(1741) – Lagrange (1777), or employing <strong>the</strong> technique of adjoint-exact <strong>linear</strong> <strong>differential</strong><br />
operator .<br />
Here we c<strong>on</strong>sider <strong>the</strong> <strong>differential</strong> equati<strong>on</strong> of third <strong>order</strong>:<br />
(5)<br />
and we accept <strong>the</strong> knowledge of <strong>the</strong> soluti<strong>on</strong>s & of its HE:<br />
(6)<br />
with <strong>the</strong> aim to find expressi<strong>on</strong>s for <strong>the</strong> particular soluti<strong>on</strong> of (5) and <strong>the</strong> soluti<strong>on</strong> of (6).<br />
THIRD ORDER LINEAR DIFFERENTIAL EQUATION<br />
In this case, <strong>the</strong> HE (6) has three soluti<strong>on</strong>s:<br />
(7)<br />
whose <strong>linear</strong> independence implies a n<strong>on</strong>-null Wr<strong>on</strong>skian :<br />
W (8)<br />
The derivative of (8) gives:<br />
, (9)<br />
with <strong>the</strong> notati<strong>on</strong>:<br />
(10)<br />
If (9) is multiplied by u(x) and we use (7), <strong>the</strong>n:<br />
8
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