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(y′′)2+6ety′=4t

2 n d 2^{nd} 2 n d order, nonlinear

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

we're asked to state the order of the differential equation and indicate whether it is linear or nonlinear. The differential equation we're given here is y double prime squared plus six e to the power of t times y prime equals four t. Remember, we determined the order based on the highest derivative present in our equation. Here we have the second derivative indicated by that y double prime and the first derivative given by y prime. So this makes this a second order differential equation based on that highest order derivative being the second one. Now, in order to determine whether this is linear or nonlinear, remember there's three things that we check for. If any one of those three things are not true, we know that our equation isn't linear. Now one of the things that we look for is that y and its derivatives are only raised to the first power. The first thing I notice in this equation is that y double prime, my second derivative, is being squared, making this not a linear differential equation. So this is therefore a second order nonlinear differential equation. Let us know if you have any questions here and I'll see you in the next one.

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

2 n d 2^{nd} 2 n d order, nonlinear

2 n d 2^{nd} 2 n d order, linear

1 s t 1^{st} 1 s t order, linear

1 s t 1^{st} 1 s t order, nonlinear

13: Intro to Differential Equations

Basics of Differential Equations

Business Calculus

13: Intro to Differential Equations

Basics of Differential Equations

Struggling with Business Calculus?

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Multiple Choice

State the order of the differential equation and indicate if it is linear or nonlinear.

$\left(y^{\prime\prime}\right)^2+6e^{t}y^{\prime}=4t$

$2^{nd}$ order, linear

$2^{nd}$ order, nonlinear

$1^{st}$ order, linear

$1^{st}$ order, nonlinear

Verified step by step guidance

Step 1: Identify the highest derivative present in the given differential equation. The equation is $(y^{\prime\prime})^2 + 6e^t y^{\prime} = 4t$. The highest derivative is $y^{\prime\prime}$, which is the second derivative of $y$. Therefore, the equation is of second order.

Step 2: Determine if the equation is linear or nonlinear. A differential equation is linear if all terms involving the dependent variable $y$ and its derivatives $y^{\prime}, y^{\prime\prime}, \dots$ appear to the first power and are not multiplied together. In this equation, $(y^{\prime\prime})^2$ involves the second derivative squared, which makes the equation nonlinear.

Step 3: Verify the linearity condition for the remaining terms. The term $6e^t y^{\prime}$ is linear because $y^{\prime}$ appears to the first power and is not multiplied by another derivative or $y$. However, the presence of $(y^{\prime\prime})^2$ overrides this and confirms nonlinearity.

Step 4: Summarize the findings. The equation is of second order because the highest derivative is $y^{\prime\prime}$, and it is nonlinear because of the $(y^{\prime\prime})^2$ term.

Step 5: Conclude that the correct classification of the differential equation is: second order, nonlinear.

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