Question

The order and degree of differential equation (d²y/dx²)³ + (dy/dx)² + y = 0 respectively

• A. 3, 2
• B. 2, 3
• C. 2, 2
• D. 3, 1

Hint:

Always find the Order first. Once you identify the "boss" (the highest derivative), the Degree is simply the power that specific derivative is carrying. Ignore the powers of all lower-order derivatives.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Order is the highest derivative present in the equation. Degree is the power of that highest derivative, provided the equation is a polynomial in its derivatives.

Step 2: Detailed Explanation:
In the given equation: \[ \left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + y = 0 \] 1. The derivatives present are $\frac{d^2y}{dx^2}$ (second order) and $\frac{dy}{dx}$ (first order). The highest order is 2. 2. The power raised to this highest derivative ($\frac{d^2y}{dx^2}$) is 3. Thus, Order = 2 and Degree = 3.

Step 3: Final Answer:
The order and degree are 2 and 3 respectively.

Answer 2

The equation given is \( \left(\dfrac{d^2y}{dx^2}\right)^3 + \left(\dfrac{dy}{dx}\right)^2 + y = 0 \). Order is the highest derivative appearing anywhere in the equation, and degree is the power of that highest-order derivative once the equation is written as a polynomial in derivatives, with no fractional or negative powers, and no derivative inside another function like a sine or square root.

1. 3, 2: This would be correct only if the highest derivative present were a third-order derivative, but the equation contains only \(\dfrac{d^2y}{dx^2}\) and \(\dfrac{dy}{dx}\), the highest being second-order, not third, so the order given here is wrong.
2. 2, 3: The highest derivative present is \(\dfrac{d^2y}{dx^2}\), a second-order derivative, giving order 2. That same term is raised to the power 3 in the equation, giving degree 3. Both values line up with what's actually written.
3. 2, 2: The order value of 2 here is correct, since \(\dfrac{d^2y}{dx^2}\) is indeed the highest derivative, but the degree is misread as 2 instead of the actual power of 3 that the highest-order term is raised to.
4. 3, 1: Neither value matches, the order is not 3 since no third derivative appears, and even if it were, a degree of 1 doesn't correspond to the cube on the second-derivative term.

Reading the powers directly off the equation, the highest derivative is second order, and it appears cubed.

Therefore, the correct answer is 2, 3.