Question

The degree and order of the differential equation \[ \left( \frac{d^2 y}{dx^2} \right)^{\frac{4}{5}} = 10 \frac{dy}{dx} + 2 \] are:

• A. Degree 2, Order 5
• B. Degree 5, Order 1
• C. Degree 20, Order 2
• D. Degree 4, Order 2

Answer 1

The Correct Option is D

The given differential equation is \(\left( \frac{d^2 y}{dx^2} \right)^{\frac{4}{5}} = 10 \frac{dy}{dx} + 2\). To find its degree and order, we proceed as follows:

1. Order: The order is determined by the highest derivative present. In this equation, the highest derivative is \(\frac{d^2 y}{dx^2}\), the second derivative. Thus, the order is 2.
2. Degree: The degree is the highest power of the highest order derivative after eliminating fractional and negative powers. The initial equation is \(\left( \frac{d^2 y}{dx^2} \right)^{\frac{4}{5}} = 10 \frac{dy}{dx} + 2\).

To obtain a polynomial form, we raise both sides to the power of 5 to remove the fractional exponent:
\(\left( \left( \frac{d^2 y}{dx^2} \right)^{\frac{4}{5}} \right)^5 = (10 \frac{dy}{dx} + 2)^5\).
This simplifies to:
\(\left( \frac{d^2 y}{dx^2} \right)^4 = (10 \frac{dy}{dx} + 2)^5\).

The degree is now identified as the highest power of \(\frac{d^2 y}{dx^2}\), which is 4.

Consequently, the degree is 4 and the order is 2.

Attribute	Value
Degree	4
Order	2

The determined attributes are: Degree 4, Order 2.