= Solution
The origin is a <regular singular point> of this <Kummer differential equation>. Apply the <Frobenius method>, writing $y=\sum_{n=0}^\infty a_nx^{n+r}$ with $a_0\ne0$. The lowest power gives the <indicial equation>
$$
r(r+c-1)=0,\qquad r=0,\ 1-c.
$$
For $n\geq1$, comparison of the coefficient of $x^{n+r-1}$ gives
$$
(n+r)(n+r+c-1)a_n-(n+r)a_{n-1}=0,\qquad
a_n=\frac{a_{n-1}}{n+r+c-1}.
$$
Since $0<c<1$, both roots give valid independent branches. Normalize $a_0=1$:
$$
\boxed{y_1(x)=\sum_{n=0}^\infty\frac{\Gamma(c)}{\Gamma(c+n)}x^n
=\sum_{n=0}^\infty\frac{x^n}{(c)_n},\qquad
y_2(x)=x^{1-c}\sum_{n=0}^\infty\frac{x^n}{n!}=x^{1-c}e^x.}
$$
Here $(c)_n$ is the <rising factorial>, and the first series is the <confluent hypergeometric function of the first kind> $M(1,c,x)$. The two leading powers are distinct, proving <linear independence>. We describe real solutions on $x>0$, with their one-sided behavior at zero.
For the inhomogeneous equation a particular solution is $-(x+c)/2$, verified by direct substitution. Thus $y=Ay_1+By_2-(x+c)/2$. Since $y_2'\sim(1-c)x^{-c}$, a finite derivative at zero forces $B=0$. Then $y(0)=0$ forces $A=c/2$, and the required solution is
$$
\boxed{y(x)=\frac c2\left[M(1,c,x)-1-\frac xc\right]
=\frac c2\sum_{n=2}^\infty\frac{x^n}{(c)_n}.}
$$
In particular $y'(0)=0$. The singular homogeneous branch cannot be added without violating the derivative condition.
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