Euler Differential Equation

The general nonhomogeneous differential equation is given by

WIDTH="199" HEIGHT="41" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3079.gif">

(1)

and the homogeneous equation is

WIDTH="152" HEIGHT="27" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3080.gif">

(2)

WIDTH="144" HEIGHT="39" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3081.gif">

(3)

Now attempt to convert the equation from

WIDTH="164" HEIGHT="28" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3082.gif">

(4)

to one with constant coefficients

WIDTH="148" HEIGHT="41" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3083.gif">

(5)

by using the standard transformation for linear second-order ordinary differential equations. Comparing (3) and (5), the functions p(x) and q(x) are

WIDTH="121" HEIGHT="35" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3085.gif">

(6)

WIDTH="132" HEIGHT="39" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3086.gif">

(7)

Let WIDTH="48" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="http://mathworld.wolfram.com/eimg3087.gif"> and define

WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg1073.gif">	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg397.gif">	WIDTH="287" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3088.gif">	(8)
	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg21.gif">	WIDTH="120" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3089.gif">	(9)

Then A is given by

WIDTH="16" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg930.gif">	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg397.gif">	WIDTH="160" HEIGHT="53" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3090.gif">	(10)
	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg21.gif">	WIDTH="219" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3091.gif">	(11)
	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg21.gif">	WIDTH="46" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3092.gif">	(12)

which is a constant. Therefore, the equation becomes a second-order ordinary differential equation with constant
coefficients

WIDTH="186" HEIGHT="41" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3093.gif">

(13)

Define

WIDTH="18" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3094.gif">	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg397.gif">	WIDTH="158" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3095.gif">	(14)
	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg21.gif">	WIDTH="202" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3096.gif">	(15)
WIDTH="18" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3097.gif">	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg397.gif">	WIDTH="158" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3098.gif">	(16)
	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg21.gif">	WIDTH="202" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3099.gif">	(17)

and

WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg2316.gif">	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg397.gif">	WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3100.gif">	(18)
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg863.gif">	WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg397.gif">	WIDTH="129" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3101.gif">	(19)

The solutions are

WIDTH="341" HEIGHT="64" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3102.gif">

(20)

In terms of the original variable x,

WIDTH="405" HEIGHT="64" BORDER="0" SRC="http://mathworld.wolfram.com/eimg3103.gif">

(21)

Posted by jongwook