State the order of the differential equation and indicate if it is linear or nonlinear. 
$y^{{}^{\prime \prime \prime }}+3xy=4\sqrt{x}$

• A. $3^{rd}$ order; nonlinear
• B. $1^{st}$ order; linear
• C. $3^{rd}$ order; linear
• D. $1^{st}$ order; nonlinear

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem. A differential equation's order is determined by the highest derivative present in the equation. Additionally, the equation is classified as linear if it can be written in the form where the dependent variable and its derivatives appear to the first power and are not multiplied together.
2. Step 2: Analyze the given equation. The equation provided is y′′′ + 3xy = 4√x. Here, y′′′ (the third derivative of y) is the highest derivative present, so the order of the differential equation is 3rd (third).
3. Step 3: Check for linearity. For a differential equation to be linear, the dependent variable (y) and its derivatives (y′, y′′, y′′′, etc.) must appear to the first power and cannot be multiplied by each other. In this equation, y′′′, y, and the term 3xy all satisfy this condition, as there are no powers or products of y and its derivatives.
4. Step 4: Confirm the classification. Since the equation is of the 3rd order and satisfies the conditions for linearity, it is classified as a 3rd order linear differential equation.
5. Step 5: Finalize the answer. The correct classification of the given differential equation is: 3rd order; linear.