Question

Find all the values of r such that y=Ax^r satisfies the equation:
y''+(x^-1)y'-y^3=0

Give a reason why other solutions must exist for this equation.

Answer 1

y=x^r
y'=rx^r-1
y''=r(r-1)x^r-2
r(r-1)x^r-2+x^-1rx^-1-(y^3)=0
r(r-1)x^-2+rx^-2-y^3=0

do you know how to get y^3 in terms or r?

Answer 2

r^2-r+r=y^3
r^2=x^5 because divide everything by x^-2
and im taking it as A is a constant
r^2=Ax^5
2ln(r)=5ln(x)
ln(r)=5/2ln(x)
r=e^x^5/2

but i dont know i never did this one in linear algebra/D.E's