State the order of the differential equation and indicate if it is linear or nonlinear.
${(y^{''})}^{2} + 6 e^{t} y^{'} = 4 t$

$2 raised to the n d power$ order, linear

$2^{nd}$ order, nonlinear

$1 raised to the s t power$ order, linear

$1^{st}$ order, nonlinear

Verified step by step guidance

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.

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.

Conclude that the differential equation is of second order because the highest derivative is $y^{\prime\prime}$.

Conclude that the differential equation is nonlinear because $(y^{\prime\prime})^2$ introduces a nonlinearity.

Summarize: The differential equation is a second-order, nonlinear equation.

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

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State the order of the differential equation and indicate if it is linear or nonlinear.
${(y^{''})}^{2} + 6 e^{t} y^{'} = 4 t$