What is the solution of a partial differential equation

Ordinary differential equations have two classes of solutions —the complete integral and the singular solution. Particular solutions are only varieties of the complete integral. Three 

When a and b are arbitrary constants, each or both of which may have any assigned magnitude, equation (2) may represent two infinite systems of spheres of radius r. The centre of any member of either of these two infinite systems—called a double infinite system—must lie somewhere on the xy-plane. 

Equation (2), therefore, is the complete integral of (4). By assigning any particular numerical value to a or b, a particular solution of (4) will be obtained, such is 

This result satisfies equation (4), but, unlike the particular solution, is not included in the complete integral (2). Such a solution of the differential equation is said to be a singular solution. 

Geometrically, the singular solution represents two plane surfaces touched by all the spheres represented by equation (2). The singular.solution is thus the envelope of all the spheres represented 

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