General solution to a differential equation with an initial condition How can I find the general solution of this DE? where $D$, $R$, and $I$ are constant parameters.  
$$\frac{D}{2}\frac{\ d ^2 y}{\ d x^2 \,} + {(x-RI-D)}\frac{\ d y}{\ d x} + (1+\lambda){y}{} = 0$$  
and the initial conditions:  
$$y(x_0)=a\space;\dot y(x_0)=b$$   
 A: 
I would like to answer this question just for mere curiosity.

Let me write your ODE as follows:
$$ y'' + \beta(x-\alpha) y' + \mu y = 0, \quad \beta = 2/D, \quad \alpha = RI+D, \quad  \mu = 2(1+\lambda)/D. $$
Notice that every $x \in \mathbb{R}$ is a regular point of your equation and then we can expand the solution about $x=0$ (for example). Thus, we have, assuming uniform convergence:
$$y(x) = \sum^\infty_{n=0} c_n x^n, \quad y'(x) = \sum^\infty_{n=0} nc_n x^{n-1}, \quad y''(x) = \sum^\infty_{n=0} n (n-1) c_n x^{n-2}, \quad $$
which after substitution leads to:
$$ \sum^\infty_{n=0} n(n-1)c_n x^{n-2}- \alpha \beta \sum^\infty_{n=0} n c_n x^{n-1} + \sum_{n=0}^\infty (\mu +\beta n)c_nx^n = 0,$$
perform a change of indices in the two first series to have:
$$\sum^\infty_{n=-2} (n+2)(n+1)c_{n+2}x^n - \alpha \beta \sum^\infty_{n=-1}(n+1)c_{n+1}x^n + \sum_{n=0}^\infty (\mu +\beta n)c_nx^n = 0 .$$
Setting all the coefficients of the expression above to $0$, we will come up with:
\begin{align}
n = -2: & \quad 0 \cdot c_0 \cdot x^0 = 0,  \\
n = -1: & \quad  0 \cdot c_1 \cdot x^1 - \alpha \beta \cdot c_0 \cdot 0  = 0  \\
n \geq 0: & \quad c_{n+2} = \frac{\alpha \beta (n+1)c_{n+1} - (\mu + \beta n )c_n}{(n+2)(n+1)}.
\end{align}
The last expression gives you the general term for the series as a function of $c_0$ and $c_1$, provided they are $\neq 0$. This will give you the two parts of the homogenous solution, which in your case is the full solution, in form of two separate series. Substitute your initial data and discuss the result as a function of $\lambda$. 
Mathematica seems to give the following closed result for your ODE:

$$\color{blue}{\begin{align}
y(x) = & A \, e^{-x^2 \beta / 2 + \alpha \beta x} H_{\frac{\mu}{\beta}-1}\left( \frac{x \sqrt{\beta }}{\sqrt{2}}-\frac{\alpha  \sqrt{\beta }}{\sqrt{2}}\right) + \\
       & B \, e^{-x^2 \beta / 2 + \alpha \beta x} {}_1F_1\left( \frac{\beta -\mu }{2 \beta },\frac{1}{2},\left(\frac{x \sqrt{\beta }}{\sqrt{2}}-\frac{\alpha  \sqrt{\beta }}{\sqrt{2}}\right)^2\right),
\end{align}}$$

where $H_n(x)$ is the Hermite polynomial of order $n$ and ${}_1F_1(a,b,x)$ is the Kummer confluent hypergeometric function.

Hope somebody finds this helpful.
Cheers!
