Question

Match List I with List II

List I		List II
A.	\(\frac{d^2y}{dx^2}+(\frac{dy}{dx})^{\frac{1}{2}}+x^{\frac{1}{2}}\)	I.	order 2, degree 1
B.	\(\frac{dy}{dx}=\frac{x^{\frac{1}{2}}}{y^{\frac{1}{2}}(1+x)^{\frac{1}{2}}}\)	II.	order 2, degree not defined
C.	\(\frac{d^2y}{dx^2}=\cos3x+\sin3x\)	III.	order 2, degree 4
D.	\(\frac{d^2y}{dx^2}+2\frac{dy}{dx}+y=\log(\frac{dy}{dx})\)	IV.	order 1, degree 2

Choose the correct answer from the options given below :

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

Answer 1

The Correct Option is B

To solve the problem of matching equations in List I with their respective orders and degrees in List II, we need to analyze each differential equation:

• Equation A: \(\frac{d^2y}{dx^2}+(\frac{dy}{dx})^{\frac{1}{2}}+x^{\frac{1}{2}}\)
  • Order: 2, as the highest derivative is \(\frac{d^2y}{dx^2}\).
  • Degree is not defined because \((\frac{dy}{dx})^{\frac{1}{2}}\) involves a fractional power of a derivative.
• Equation B: \(\frac{dy}{dx}=\frac{x^{\frac{1}{2}}}{y^{\frac{1}{2}}(1+x)^{\frac{1}{2}}}\)
  • Order: 1, as the highest order derivative is \(\frac{dy}{dx}\).
  • Degree: 2, since the equation can be expressed without any fractional powers on the derivative, after some manipulation.
• Equation C: \(\frac{d^2y}{dx^2}=\cos3x+\sin3x\)
  • Order: 2, as the highest derivative is \(\frac{d^2y}{dx^2}\).
  • Degree: 1, because all terms on the right don't involve derivatives.
• Equation D: \(\frac{d^2y}{dx^2}+2\frac{dy}{dx}+y=\log(\frac{dy}{dx})\)
  • Order: 2, as the highest derivative is \(\frac{d^2y}{dx^2}\).
  • Degree is not defined because \(\log(\frac{dy}{dx})\) is a non-polynomial term involving a derivative.

Matching the characteristics with List II gives us:

• A: Order 2, Degree not defined - Matches with III.
• B: Order 1, Degree 2 - Matches with IV.
• C: Order 2, Degree 1 - Matches with I.
• D: Order 2, Degree not defined - Matches with II.

Thus, the correct mapping is: A-III, B-IV, C-I, D-II. Therefore, the correct answer is:

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