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: Definitions:

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

The degree is the power of the highest-order derivative after removing radicals and fractions involving derivatives.

A differential equation is linear if the dependent variable and its derivatives appear only to the first power and are not part of other functions (like \( \sin(y) \)) or multiplied together.

Step 2: Examples:

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

• Highest derivative: \( \frac{d^2y}{dx^2} \), so Order = 2.
• Raise both sides to the power of 3: \( \left( y + x \left( \frac{dy}{dx} \right)^2 \right)^5 = x^3 \left( \frac{d^2y}{dx^2} \right)^3 \).
• Degree of 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: \( \frac{d^2y}{dx^2} \), so Order = 2.
• Raise both sides to the power of 6: \( \left( \frac{d^2y}{dx^2} \right)^2 = \left( y + \frac{dy}{dx} \right)^3 \).
• Degree of 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: \( \frac{dy}{dx} \), so Order = 1.
• Isolate radical and square both sides: \( \left( y - x \frac{dy}{dx} \right)^2 = 1 + \left( \frac{dy}{dx} \right)^2 \).
• After simplification, 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: \( (2 + x^3) \frac{dy}{dx} - y = \sqrt{e^{\sin x}} \).
• Highest derivative: \( \frac{dy}{dx} \), so Order = 1.
• The dependent variable \( y \) and its derivative \( y' \) appear to the first power. Coefficients are functions of \( x \).
• The degree is 1 and the equation is linear.
• Match: D - II.

Step 3: Answer:

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