Singular Solution of a Differential Equation

The given statement is true. A singular solution of a differential equation is one that cannot be obtained from the general solution obtained by the usual method of solving the differential equation.

General Solution
The general solution of a differential equation contains all possible solutions to the equation. It is obtained by integrating the differential equation and including an arbitrary constant.

Method of Solving Differential Equations
The usual method of solving a differential equation involves finding the general solution by integrating the equation and then applying initial or boundary conditions to determine the particular solution.

Singular Solution
A singular solution is a special solution that cannot be derived from the general solution. It arises when the arbitrary constant in the general solution takes on a specific value that satisfies additional conditions or constraints. These additional conditions are not accounted for in the general solution.

Example
Consider the differential equation: dy/dx = x/y

The general solution of this differential equation can be obtained by separating the variables and integrating both sides:
∫y dy = ∫x dx
y^2/2 = x^2/2 + C

Here, C is the arbitrary constant and represents the general solution.

However, suppose we have an additional condition that the solution must pass through the point (1, 1). Substituting these values into the general solution, we get:
(1)^2/2 = (1)^2/2 + C
1/2 = 1/2 + C
C = 0

The singular solution is obtained when the arbitrary constant C takes on a specific value of zero. Therefore, the singular solution to the differential equation is:
y^2/2 = x^2/2

This singular solution satisfies the additional condition and cannot be obtained from the general solution.

Conclusion
In conclusion, a singular solution of a differential equation is one that arises when the arbitrary constant in the general solution takes on a specific value that satisfies additional conditions or constraints. These singular solutions cannot be obtained from the general solution obtained by the usual method of solving the differential equation.