Question

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

LIST-I (Differential)	LIST-II (Order/degree / nature)
(A) \( \left(y + x\left(\frac{dy}{dx}\right)^2\right)^{5/3} = x \frac{d^2y}{dx^2} \)	(I) order = 2, degree = 2, non-linear
(B) \( \left(\frac{d^2y}{dx^2}\right)^{1/3} = \left(y + \frac{dy}{dx}\right)^{1/2} \)	(III) order = 2, degree = 3, non-linear
(C) \( y = x \frac{dy}{dx} + \left[1 + \left(\frac{dy}{dx}\right)^2\right]^{1/2} \)	(IV) order = 1, degree = 2, non-linear
(D) \( (2 + x^3) \frac{dy}{dx} = \left(e^{\sin x}\right)^{1/2} + y \)	(II) order = 1, degree = 1, linear

Choose the correct answer from the options given below:

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

Hint:

To find the degree of a differential equation, you must first make the equation a polynomial in its derivatives. This means eliminating all fractional powers and radicals involving any derivative terms. The highest power of the highest-order derivative in the resulting polynomial equation is the degree.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept: 

Order of a differential equation is the order of the highest derivative present.

Degree is the highest power of the highest order derivative, after the equation has been cleared of radicals and fractions in its derivatives.

Linearity: An equation is linear if the dependent variable and its derivatives appear only to the first power and are not part of any other function (like \( \sin(y) \)) or multiplied together.

Step 2: Detailed Explanation:

A. \( \left( y + x \left( \frac{dy}{dx} \right)^2 \right)^{5/3} = x \frac{d^2y}{dx^2} \):

• Highest derivative is \( \frac{d^2y}{dx^2} \), so Order = 2.
• To find the degree, cube both sides: \( \left( y + x \left( \frac{dy}{dx} \right)^2 \right)^5 = x^3 \left( \frac{d^2y}{dx^2} \right)^3 \).
• The power of the highest derivative (\( y'' \)) is 3, so Degree = 3.
• The equation is non-linear.
• Match: A - III.

B. \( \left( \frac{d^2y}{dx^2} \right)^{1/3} = \left( y + \frac{dy}{dx} \right)^{1/2} \):

• Highest derivative is \( \frac{d^2y}{dx^2} \), so Order = 2.
• To clear radicals, raise both sides to the power of 6 (LCM of 3 and 2): \( \left( \frac{d^2y}{dx^2} \right)^2 = \left( y + \frac{dy}{dx} \right)^3 \).
• The power of the highest derivative (\( y'' \)) is 2, so Degree = 2.
• The equation is non-linear.
• Match: B - I.

C. \( y = x \frac{dy}{dx} + \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \):

• Highest derivative is \( \frac{dy}{dx} \), so Order = 1.
• Isolate the radical and square both sides: \( \left( y - x \frac{dy}{dx} \right)^2 = 1 + \left( \frac{dy}{dx} \right)^2 \).
• After simplifying, the highest power of \( y' \) is 2, so Degree = 2.
• The equation is non-linear.
• Match: C - IV.

D. \( (2 + x^3) \frac{dy}{dx} = (e^{\sin x})^{1/2} + y \):

• Rearrange to standard form: \( (2 + x^3) \frac{dy}{dx} - y = \sqrt{e^{\sin x}} \).
• Highest derivative is \( \frac{dy}{dx} \), so Order = 1.
• The dependent variable \( y \) and its derivative \( y' \) appear only to the first power. The coefficients are functions of \( x \).
• The degree is 1 and the equation is linear.
• Match: D - II.

Step 3: Final Answer:

The correct matching is A-III, B-I, C-IV, D-II, which corresponds to option (C).