Question

The order and degree of the differential equation \[ \left(\frac{d^2y}{dx^2}\right)^3+\left(\frac{dy}{dx}\right)^2+y=0 \] are respectively:

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

Hint:

Order depends only on the highest derivative present, whereas degree depends on the power of that highest derivative after simplifying the equation into polynomial form.

Answer 1

The Correct Option is A

Concept:
• The order of a differential equation is the order of the highest derivative present.
• The degree is the power of the highest order derivative after the equation is free from radicals and fractional powers.

Step 1: Identify the highest order derivative.
The equation is: \[ \left(\frac{d^2y}{dx^2}\right)^3+\left(\frac{dy}{dx}\right)^2+y=0 \] The highest derivative present is: \[ \frac{d^2y}{dx^2} \] Hence, \[ \text{Order}=2 \]

Step 2: Determine the degree.
The highest order derivative appears as: \[ \left(\frac{d^2y}{dx^2}\right)^3 \] The power is: \[ 3 \] Therefore, \[ \text{Degree}=3 \] Hence, \[ \boxed{(2,3)} \]