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

$3^{rd}$ order; nonlinear

$1^{st}$ order; linear

$3^{rd}$ order; linear

$1^{st}$ order; nonlinear

Verified step by step guidance

Step 1: Understand the order of a differential equation. The order is determined by the highest derivative present in the equation. For example, if the equation contains y′′′ (the third derivative of y), the order is 3.

Step 2: Examine the given equation: y′′′ + 3xy = 4√x. Identify the highest derivative present in the equation. In this case, y′′′ (third derivative of y) is the highest derivative, so the order is 3.

Step 3: Determine if the equation is linear or nonlinear. A differential equation is linear if the dependent variable (y) and its derivatives appear to the first power and are not multiplied by each other. Otherwise, it is nonlinear.

Step 4: Analyze the given equation: y′′′ + 3xy = 4√x. Here, y′′′ and y appear to the first power, and there are no products of y and its derivatives. Therefore, the equation is linear.

Step 5: Conclude that the differential equation is of 3rd order and linear based on the analysis of the highest derivative and the linearity criteria.

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

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