**Differential Equations: Order and Linearity**1. **Determine the order of the given differential equation; also state whether the equation is linear or nonlinear**: (a) $ x^2 \frac{d^4 y}{dx^4} + (y - 1) \frac{dy}{dx} + 4y = \cos x $ **Explanation**: - The highest derivative present in the equation is $\frac{d^4 y}{dx^4}$, hence the order of the differential equation is 4. - The equation is nonlinear because the term $(y - 1) \frac{dy}{dx}$ involves a product of $y$ and its derivative, making it nonlinear. (b) $ dy + (xy - \cos x) dx = 0 $ **Explanation**: - This can be rewritten as $\frac{dy}{dx} + xy = \cos x$, which shows that the highest derivative is $\frac{dy}{dx}$. Therefore, the order of the differential equation is 1. - The equation is linear because it can be expressed in the form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x) = x$ and $Q(x) = \cos x$, indicating linearity.

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Answered: 1. Determine the order of the given…

1. Determine the order of the given differential equation; also state whether the equation is linear or nonlinear. (a) r². 2 dªy + (y – 1). dr4 dy + 4y = cos x dr (b) dy + (xy – cos a)dr = 0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

**Differential Equations: Order and Linearity**

1. **Determine the order of the given differential equation; also state whether the equation is linear or nonlinear**:

(a) $ x^2 \frac{d^4 y}{dx^4} + (y - 1) \frac{dy}{dx} + 4y = \cos x $

**Explanation**:
- The highest derivative present in the equation is $\frac{d^4 y}{dx^4}$, hence the order of the differential equation is 4.
- The equation is nonlinear because the term $(y - 1) \frac{dy}{dx}$ involves a product of $y$ and its derivative, making it nonlinear.

(b) $ dy + (xy - \cos x) dx = 0 $

**Explanation**:
- This can be rewritten as $\frac{dy}{dx} + xy = \cos x$, which shows that the highest derivative is $\frac{dy}{dx}$. Therefore, the order of the differential equation is 1.
- The equation is linear because it can be expressed in the form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x) = x$ and $Q(x) = \cos x$, indicating linearity.

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