Question

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

• A. \( 1, 1 \)
• B. \( 1, 2 \)
• C. \( 1, 3 \)
• D. \( 2, 1 \)
• E. \( 2, 2 \)

Hint:

To find the degree, always ensure all derivatives are expressed as powers in a polynomial. If you see a square root or a fractional power on a derivative, you must rationalize the equation first.

Answer 1

Answer: (E)

Concept:
• Order: The order of a differential equation is the order of the highest derivative present in the equation.
• Degree: The degree is the power of the highest order derivative, provided the equation is in a polynomial form with respect to its derivatives (free from radicals and fractions).

Step 1: Determine the order.
The highest derivative in the equation is \( \frac{d^2y}{dx^2} \). Therefore, the order is 2.

Step 2: Remove radicals to find the degree.
The term \( \left( \frac{dy}{dx} \right)^{\frac{3}{2}} \) has a fractional power. We must isolate the radical and square the equation: \[ \frac{d^2y}{dx^2} - y = -\left( \frac{dy}{dx} \right)^{\frac{3}{2}} \] Squaring both sides: \[ \left( \frac{d^2y}{dx^2} - y \right)^2 = \left( \frac{dy}{dx} \right)^3 \] \[ \left( \frac{d^2y}{dx^2} \right)^2 - 2y \frac{d^2y}{dx^2} + y^2 = \left( \frac{dy}{dx} \right)^3 \]

Step 3: Determine the degree.
The highest order derivative is \( \frac{d^2y}{dx^2} \), and its power in the polynomial form is 2. Therefore, the degree is 2.