State the order of the differential equation and indicate if it is linear or nonlinear. 
${\left(y^{\prime \prime }\right)}^2+6e^t{y}^{\prime }=4t$

• A. $2^{nd}$ order, linear
• B. $2^{nd}$ order, nonlinear
• C. $1^{st}$ order, linear
• D. $1^{st}$ order, nonlinear

Answer: (B)

Verified step by step guidance

1. Identify the highest derivative present in the given differential equation. The equation is \((y^{\prime\prime})^2 + 6e^t y^{\prime} = 4t\), and the highest derivative is \(y^{\prime\prime}\), which is the second derivative. Therefore, the equation is of second order.
2. 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 together. In this equation, \((y^{\prime\prime})^2\) involves the square of the second derivative, making it nonlinear.
3. Conclude that the differential equation is of second order because the highest derivative is \(y^{\prime\prime}\).
4. Conclude that the differential equation is nonlinear because \((y^{\prime\prime})^2\) introduces a nonlinearity.
5. Summarize: The differential equation is a second-order, nonlinear equation.