Solving differential equation (Numerical & Analytical)? I want to solve the following differential equation

$y''(x)\ /\ y(x)= \frac{\lambda\ x^{\frac{3}{4}}}{\sqrt{1 - x}}\ ,\ 0\lt x\lt 1$

But do not know how to actually solve it. Any suggestion?
 A: The second order linear ODE can be written as follows:
$$
y''(x)- \frac{\lambda x^{\frac{3}{4}}}{\sqrt{1-x}}y(x)=0
$$
To solve this numerically, one can employ second order finite difference as follows. Consider $N$ points equally spaced in $[0,1]$ i.e $(0,x_{1},x_{2},...,x_{N},1)$. We can approximate $y''(x_{j})$as
$$
y''(x_{j})\approx \frac{y(x_{j+1})-2y(x_{j})+y(x_{j-1})}{h^2}
$$
Note $h=x_{n}-x_{n-1}$. With this, the discrete scheme can be written as:
$$
\frac{y(x_{j+1})-2y(x_{j})+y(x_{j-1})}{h^2}-\frac{\lambda x_j^{\frac{3}{4}}}{\sqrt{1-x_j}}y(x_j)=0
$$
Now consider the above equation for $j=0,1,2,...N$. We get a tridiagonal matrix(verify!) and the following linear algebra problem.
$$
\left[A-KI\right]{\bf y}=0
$$
where 
$$K=\lambda \frac{\bf{x}^{\frac{3}{4}}}{\sqrt{1-\bf{x}}}$$
$y$ is the vector of unknowns, namely $y(x_{1}),y(x_{2})...y(x_{N})$. You need to incorporate boundary conditions. This will result a slight modification of the linear system.
You can write a code to solve this with your choice of programming language. Choose a suitable value of $h$("small") and take "large" $N$ as you desire.
