Example 4.  Investigate the initial value problem    with  .   

Solution 4.

The function    has a removable singularity at  ,  because it's series expansion has a simple zero at  .   

Therefore, the function    has a simple pole at  .   

The solution    to the D. E.     has a movable singularity at the point .  

Therefore, the differential equation    has the Painlevé property.   

(c) John H. Mathews 2005