A singular solution of the given differential equation isa)Either a general solution or a particular solutionb)Neither a general solution nor a particular solution but contains the arbitrary constantsc)Neither a general solution nor a particular solution and does not contain any arbitrary constantd)Known as a general solutionCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Singular Solution of a Differential Equation

A singular solution of a differential equation is a solution that cannot be obtained from the general solution by assigning specific values to the arbitrary constants. In other words, it is a solution that does not contain any arbitrary constants and cannot be expressed in terms of the general solution.

Explanation:

1. General Solution:
The general solution of a differential equation is a solution that contains arbitrary constants. It represents a family of solutions that can be obtained by assigning different values to the arbitrary constants.

2. Particular Solution:
A particular solution of a differential equation is a solution that is obtained by assigning specific values to the arbitrary constants in the general solution. It represents a specific solution for a given set of conditions or initial values.

3. Singular Solution:
A singular solution of a differential equation is a solution that neither contains arbitrary constants nor can be obtained from the general solution by assigning specific values to the arbitrary constants.

Example:
Consider the differential equation dy/dx = x.
The general solution of this differential equation is y = (1/2)x^2 + C, where C is an arbitrary constant.

If we assign a specific value to C, say C = 1, we obtain a particular solution y = (1/2)x^2 + 1. This particular solution represents a specific solution for a given set of initial conditions.

However, if we consider the equation y = x^2, it does not contain any arbitrary constant and cannot be obtained from the general solution by assigning specific values to the arbitrary constants. Hence, it is a singular solution.

Conclusion:

A singular solution of a differential equation is a solution that neither contains arbitrary constants nor can be obtained from the general solution by assigning specific values to the arbitrary constants. It represents a unique solution that does not belong to the family of solutions given by the general solution.

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