Question

Match List-I with List-II and choose the correct option:

LIST-I	LIST-II
(A) The solution of an ordinary differential equation of order 'n' has	(III) 'n' arbitrary constants
(B) The solution of a differential equation which contains no arbitrary constant is	(IV) particular solution
(C) The solution of a differential equation which is not obtained from the general solution is	(I) singular solution
(D) The solution of a differential equation containing as many arbitrary constants as the order of a differential equation is	(II) complete primitive

Choose the correct answer from the options given below:

• A. A - I, B - II, C - III, D - IV
• B. A - I, B - III, C - II, D - IV
• C. A - I, B - II, C - IV, D - III
• D. A - III, B - IV, C - I, D - II

Hint:

To remember the difference between solutions:
\textbf{General Solution / Complete Primitive:} Family of curves with 'n' constants for an nth-order ODE.
\textbf{Particular Solution:} One specific curve from the family (no constants).
\textbf{Singular Solution:} An "outsider" curve that also solves the ODE (e.g., an envelope).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question tests the fundamental definitions related to the solutions of ordinary differential equations (ODEs). It requires matching the description of a type of solution with its correct terminology.

Step 2: Detailed Explanation:
Let's analyze each item in List-I and find its corresponding definition in List-II.
A. The solution of an ordinary differential equation of order 'n' has...
The general solution of an nth-order ODE is characterized by the presence of 'n' independent arbitrary constants. These constants arise from the 'n' integrations required to solve the equation. Therefore, this description matches with III. 'n' arbitrary constants.
Match: A - III
B. The solution of a differential equation which contains no arbitrary constant is...
A particular solution is a solution obtained from the general solution by assigning specific values to the arbitrary constants. As a result, it contains no arbitrary constants. Therefore, this description matches with IV. particular solution.
Match: B - IV
C. The solution of a differential equation which is not obtained from the general solution is...
A singular solution is an exceptional solution to an ODE that cannot be obtained by specializing the arbitrary constants of the general solution. It is often an envelope to the family of curves represented by the general solution. Therefore, this description matches with I. singular solution.
Match: C - I
D. The solution of a differential equation containing as many as arbitrary constants as the order of a differential equation is...
This is the definition of the general solution of an ODE. Another name for the general solution is the "complete primitive." It represents the entire family of functions that satisfy the ODE. Therefore, this description matches with II. complete primitive.
Match: D - II

Step 3: Final Answer:
Combining the matches, we get:
A \(\to\) III
B \(\to\) IV
C \(\to\) I
D \(\to\) II This combination corresponds to option (D).