State the order of the differential equation and indicate if it is linear or nonlinear. d 2 y d x 2 − ( d y d x ) ( 1 − x ) = 2 \frac{d^2y}{dx^2}-\left(\frac{dy}{dx}\right)(1-x)=2

we're asked to state the order of the differential equation and indicate whether it is linear or nonlinear. The differential equation that we have here is d squared y over d x squared, that's my second derivative of y with respect to x, minus d y d x times one minus x is equal to two. So the first thing that we want to find here is the order of this differential equation, which remember is based on the highest derivative present in the equation. Here we can see that we have our second derivative and our first derivative making this a second order differential equation because that's my highest order derivative there. Then, to find whether this is linear or nonlinear, remember we need to check for three different things. The first thing that we want to check here is that y and its derivatives are not being multiplied by each other. I can see that my second derivative and my first derivative are in separate terms not multiplying each other, so we are all good there. The next thing that we want to check for is that y and its derivatives are only raised to the first power. Now be careful here because this two in my exponent does not mean that we are squaring this because of the specific way that it's written. This indicates a second derivative, not anything being squared. So we're all good for that term and d y d x that is also not raised to any power besides one. So we can move on to the final thing to tell us whether this is linear or nonlinear, checking that y and its derivatives are not inside of a function. So they're not in the argument of a function. I can see that that's true here as well, making this a linear differential equation. So all in all, this is a second order linear differential equation. Let us know if you have any questions, and I'll see you in the next one.

State the order of the differential equation and indicate if it is linear or nonlinear. d 2 y d x 2 − ( d y d x ) ( 1 − x ) = 2 \frac{d^2y}{dx^2}-\left(\frac{dy}{dx}\right)(1-x)=2

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

3 r d 3^{rd} 3 r d order, linear

13. Intro to Differential Equations

Basics of Differential Equations

Calculus

13. Intro to Differential Equations

Basics of Differential Equations

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Multiple Choice

State the order of the differential equation and indicate if it is linear or nonlinear.
$\frac{d^{2} y}{d x^{2}} - (\frac{d y}{d x}) (1 - x) = 2$

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

$2^{nd}$ order, nonlinear

$3 raised to the r d power$ order, nonlinear

$3^{rd}$ order, linear

Verified step by step guidance

Identify the highest derivative present in the given differential equation. The equation is $ \frac{d^2y}{dx^2} - \left(\frac{dy}{dx}\right)(1-x) = 2 $. The highest derivative is $ \frac{d^2y}{dx^2} $, which is the second derivative. Therefore, the order of the differential equation is 2.

Determine if the equation is linear or nonlinear. A differential equation is linear if the dependent variable (in this case, $ y $) and all its derivatives appear to the first power and are not multiplied by each other. In this equation, $ \frac{d^2y}{dx^2} $ and $ \frac{dy}{dx} $ appear to the first power, and there are no products of $ y $ or its derivatives. Thus, the equation is linear.

Summarize the findings: The differential equation is of the second order and is linear.

Compare the findings with the provided answer choices. The correct answer is '2nd order, linear.'

Conclude that the equation satisfies the criteria for being a second-order linear differential equation.

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Struggling with Calculus?

State the order of the differential equation and indicate if it is linear or nonlinear. d2ydx2−(dydx)(1−x)=2\frac{d^2y}{dx^2}-\left(\frac{dy}{dx}\right)(1-x)=2

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Basics of Differential Equations practice set

State the order of the differential equation and indicate if it is linear or nonlinear.
$\frac{d^{2} y}{d x^{2}} - (\frac{d y}{d x}) (1 - x) = 2$