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)=2dx2d2y−(dxdy)(1−x)=2

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

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 d x 2 d 2 y − ( d x d y ) ( 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 d x 2 d 2 y − ( d x d y ) ( 1 − x ) = 2

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

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

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

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

13: Intro to Differential Equations

Basics of Differential Equations

Business 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.

$\frac{d^2y}{dx^2}-\left(\frac{dy}{dx}\right)(1-x)=2$

$2^{nd}$ order, linear

$2^{nd}$ order, nonlinear

$3^{rd}$ order, nonlinear

$3^{rd}$ order, linear

Verified step by step guidance

Step 1: Identify the highest derivative in the given differential equation. The equation includes $ \frac{d^2y}{dx^2} $, which is the second derivative of $ y $ with respect to $ x $. Therefore, the order of the differential equation is 2.

Step 2: Recall the definition of a linear differential equation. A differential equation is linear if the dependent variable (in this case, $ y $) and its derivatives appear to the first power and are not multiplied by each other.

Step 3: Examine the terms in the equation. The term $ \frac{d^2y}{dx^2} $ is linear because it appears to the first power. However, the term $ \left( \frac{dy}{dx} \right)(1-x) $ involves $ \frac{dy}{dx} $ multiplied by $ (1-x) $, which does not violate linearity since $ (1-x) $ is a function of $ x $ and not $ y $.

Step 4: Confirm that there are no nonlinear interactions between $ y $ or its derivatives (e.g., products or powers of $ y $ or its derivatives). Since all terms meet the criteria for linearity, the equation is linear.

Step 5: Conclude that the differential equation is a second-order linear differential equation based on the analysis of its order and linearity.

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