2nd order Euler-Cauchy equation     ODE/Euler.mws   example

>

restart;

>	with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

coefficients:

>	e0:=1;e1:=3;e2:=1;

>	a0:=e0;a1:=e1-e0;a2:=e2;

>

f:=z;

auxilary equation:

>	ax:=a0*m^2+a1*m+a2;

>	solve(ax=0,m);

fundamental set:

>	y1:=exp(-z);

>	y2:=z*exp(-z);

Wronskians:

>	A:=matrix(2,2,[y1,y2,diff(y1,z),diff(y2,z)]);

>	W:=simplify(det(A));

Variation of parameter:

>	VP1:=simplify(-y2/W*f/a0);

>	VP2:=simplify(y1/W*f/a0);

>	u1:=simplify(int(VP1,z));

>	u2:=simplify(int(VP2,z));

complimentary solution:

>	yc:=c[1]*y1+c[2]*y2;

particular solution of non-homogeneous equation:

>	yp:=u1*y1+u2*y2;

>	yp:=simplify(yp);

General solution of non-homogeneous equation:

>

y:=yc+yp;

>	y:=simplify(subs(z=ln(abs(x)),y));

Solution curves of homogeneous equation:

>	p:={seq(seq(i*x+j*x*ln(abs(x))+ln(abs(x)),i=-2..2),j=-2..2)};

>	plot(p,x=-4..4,color=black);

>

>	p:={seq(seq(i*x+j*x*ln(abs(x))+ln(abs(x))+x-2,i=-2..2),j=-2..2)}:

>	plot(p,x=-2..2,color=black);

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