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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

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
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: 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.
Transcribed Image Text:**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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