The function y=x is one the solution of (x-1)y''-xy'+y=0 Solve using the reduction method to order

Ernstfalld 2020-12-02 Answered
The function y=x is one the solution of
(x1)yxy+y=0
Solve using the reduction method to order
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Expert Answer

FieniChoonin
Answered 2020-12-03 Author has 102 answers
Given y=x is one solution of (x1)yxy+y=0
Let, y=vx be another solution of the given equation.Then
(x1)(vx)x(vx)+vx=0
(x1)(v+2v)x(v+xv)+vx=0
(x1)xv+2(x1)vx2v=0
(x1)xv+(2x2x2)v=0
Now suppose v=u. On substituting
(x1)xu+(2x2x2)u=0
duu=2x2x2(x1)xdx
duu=x22x+2(x1)xdx
duu=(x1)2+1(x1)xdx
duu=x1x(1x1x1)dx
duu=(11x)dxdxx+dxx1
lnu=x2lnx+ln(x1)+lnc [c=integrating constant]
lnu=x+lnc(x1)x2
u=c(x1)x2ex
dvdx=c[1x1x2]ex
dv=c[1x1x2]exdx
dv=c[1x1x2]exdx
v=cexx+d [[f(x)+f(x)]exdx=exf(x)+c], [d=integrating constant]
Therefore
y=vx
=(cexx+d)x
=cex+dx
Therefore, another linearly independent solution is y=ex
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