So I'm getting ready for the exam by doing last years exam. It gives a DE which goes as following
$$(x^2-x)y''+(4x-2)y'+2y=0$$
This gives me:
$$\sum^\infty_{n=0}(n+r-1)(n+r)a_nx^{n+r}-\sum^\infty_{n=1}(n+r-1)(n+r-2)a_{n-1}x^{n+r}+\\4\sum^\infty_{n=0}(n+1)a_nx^{n+r}-2\sum^\infty_{n=1}(n+r-1)a_{n-1}x^{n+r}+2\sum^\infty_{n=0}a_nx^{n+r}$$
But, when finding the IE. Am I supposed to add the constants in front of the summation notation or am I supposed to leave it, like for some reason when I left it out I got the correct roots, but when I added it to the IE I got the wrong roots. The roots are supposed to be $(0,-1)$
Adding the constants:
$$((r-1)r+4r+2)a_0+\sum^\infty_{n=1}((n+r-1)(n+r)+4(n+r)+2)a_n-((n+r-2)(n+r-1)-2(n+r-1))a_{n-1}$$
Giving me the roots $(-1,-2)$
but when I don't add the constants i'll get
$$((r-1)r+r+1)a_0+\sum^\infty_{n=1}((n+r-1)(n+r)+4(n+r)+2)a_n-((n+r-2)(n+r-1)-2(n+r-1))a_{n-1}$$
which gives the roots $(0,-1)$
or is this method technically wrong and the first one should be the right one but i've managed to get lost in the algebra?
Roots of indical equations are best obtained by letting $y=x^m$ in the ODE and then by setting the coefficient of the least power of $x$. Here the least power term you get is $$(-m^2+m-2m) x^{m-1}$$. Hence, the correct indical roots are $m=0,-1$.