In this problem you will solve the differential equation

(1) Since $P(x)=$ or $Q(x)=$ are not analytic at $x=0$,
$x=$ is a singular point of the differential equation. Using Frobenius' Theorem, we must check that $xP(x)=$ and $x^2Q(x)=$ are both analytic at $x=0$. Since $xP(x)$ and $x^2Q(x)$ are analytic at $x=0$, $x=0$ is a regular singular point for the differential equation $45 x^2 y'' +7 x^2 y'+5 y =0.$ From the result of Frobenius' Theorem, we may assume that $45 x^2 y'' +7 x^2 y'+5 y =0$ has a solution of the form $y= x^r\sum\limits_{n=0}^{\infty}c_nx^n$ which converges for $x \in (0,R)$ where $r$ and $R$ are constants that will be determined later.

(2) Substituting $y = x^r\sum_{n=0}^{\infty}c_n\ x^n$ into $45 x^2 y'' +7 x^2 y'+5 y =0$, we get that

 $45 x^2$ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$ c x $+7 x^2$ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$ c x $+ 5$ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$ c x $= 0$

Multiplying the coefficients through the sums and factoring out $x^r$
 $x^r\Bigg($ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$ c x $+$ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$ c x $+$ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$ c x $\Bigg) = 0$

Reindex the sums
 $x^r\Bigg($ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$ c x $+$ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$ c x $+$ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$ c x $\Bigg) = 0$

Combining the sums ( The subscripts on the $c$'s should be increasing and numbers or in terms of $n$. )
 $x^r\Bigg($ c $+$ $\hskip 3pt\infty$$\displaystyle\sum$$n$$=$$1$ $\Bigg\lbrack$ c $+$ c $\Bigg\rbrack x^n = 0$

(3) In this step, we will use the equation above to find the indicial roots and the recurrence relation of the differential equation.

(a) From the equation above, we know that the indicial roots of the differential equation are (in increasing order) $r=$ and $r=$ .

(b) From the series above, we find that the recurrence relation is

 c $=$ c for $\geq$

(4) The general solution to $45 x^2 y'' +7 x^2 y'+5 y =0$ is

 $y$ $=$ $A$ $x$ $\Bigg($ $+$ $x +$ $x^2 +$ $x^3 + \cdots \Bigg)$ $+$ $B$ $x$ $\Bigg($ $+$ $x +$ $x^2 +$ $x^3 + \cdots \Bigg)$
and converges on the interval .
Use -inf for $-\infty$ and inf for $\infty$

Your overall score for this problem is