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?
