State the order of the differential equation and indicate if it is linear or nonlinear. 
$y^{\prime }\bullet y=3e^t$

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

Answer: (B)

Verified step by step guidance

1. Identify the highest derivative present in the differential equation. In this case, the equation is y′∙y = 3e^t, where y′ (the first derivative of y with respect to t) is the highest derivative. Therefore, the equation is a first-order differential equation.
2. Determine if the equation is linear or nonlinear. A differential equation is linear if the dependent variable (y) and its derivatives (y′, y″, etc.) appear to the first power and are not multiplied by each other. In this equation, y′ is multiplied by y, which makes it nonlinear.
3. Conclude that the differential equation is a first-order, nonlinear equation based on the analysis above.
4. Review the definitions of 'order' and 'linearity' in differential equations to reinforce understanding. The order is determined by the highest derivative, and linearity depends on whether the dependent variable and its derivatives appear linearly (not multiplied or raised to powers).
5. Verify the classification by comparing the equation to standard forms of linear and nonlinear differential equations to ensure the conclusion is correct.