Singular solution of differential equation contains
I. arbitrary constant
II. can be obtained from general
III. do not contain arbitrary constant
IV. cannot be obtained from general solution
a)
I, II are true
b)
III, IV are true
c)
I, IV are true
d)
II, IV are true
Correct answer is option 'B'. Can you explain this answer?

Question

Singular solution of differential equation contains
I. arbitrary constant
II. can be obtained from general
III. do not contain arbitrary constant
IV. cannot be obtained from general solution

a)
I, II are true
b)
III, IV are true
c)
I, IV are true
d)
II, IV are true

Correct answer is option 'B'. Can you explain this answer?

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 contains no arbitrary constants.

Options:
I. Arbitrary constant: An arbitrary constant is a constant that appears in the general solution of a differential equation and can take any value.
II. Can be obtained from general: A general solution is a solution that contains arbitrary constants and can be obtained by integrating the differential equation.
III. Do not contain arbitrary constant: A singular solution is a solution that contains no arbitrary constants.
IV. Cannot be obtained from general solution: A singular solution cannot be obtained from the general solution by assigning specific values to the arbitrary constants.

Explanation:
From the given options, we can analyze each statement:

I. Arbitrary constant: This statement is true. In the general solution of a differential equation, arbitrary constants are included to account for the different possible solutions.

II. Can be obtained from general: This statement is false. A singular solution cannot be obtained from the general solution by assigning specific values to the arbitrary constants. It is a unique solution that does not depend on the arbitrary constants.

III. Do not contain arbitrary constant: This statement is true. A singular solution is a specific solution that does not contain any arbitrary constants.

IV. Cannot be obtained from general solution: This statement is true. A singular solution is not derived from the general solution by assigning specific values to the arbitrary constants. It is a unique solution that cannot be obtained by manipulating the general solution.

Conclusion:
Based on the analysis, we can conclude that statements III and IV are true. A singular solution of a differential equation does not contain arbitrary constants and cannot be obtained from the general solution. Therefore, option B is the correct answer.

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