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: Overview:
This question assesses understanding of ordinary differential equation (ODE) solutions. It requires matching solution types to their definitions.

Step 2: Breakdown:
Analyze each item in List-I and find its corresponding definition in List-II.
A. The solution of an nth-order ODE has...
The general solution of an nth-order ODE includes 'n' independent arbitrary constants, resulting from the 'n' integrations needed to solve the equation. This aligns with III. 'n' arbitrary constants.
Match: A - III
B. The solution of a differential equation without arbitrary constants is...
A particular solution is derived from the general solution by assigning specific values to arbitrary constants, hence it lacks them. This matches IV. particular solution.
Match: B - IV
C. The solution of a differential equation not derived from the general solution is...
A singular solution is an ODE solution unattainable by specializing the general solution's arbitrary constants. It is often an envelope to the family of curves represented by the general solution. This corresponds to I. singular solution.
Match: C - I
D. The solution of a differential equation containing the same number of arbitrary constants as the equation's order is...
This defines the general solution of an ODE, also known as the "complete primitive," representing the entire family of functions satisfying the ODE. This matches II. complete primitive.
Match: D - II

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