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2. Classify each of equations given below as directly-integrable, separable, linear first- order, linear substitutive, homogeneous, Bernoulli, or none of them. Find the solu- tions to those equations. dy (1) x = 2y – 6x3 dy (2) 4x2-x2y? + 0. dx (3) x. dx dy Va = 3 dy (4) = y? – 2ary + a? dx dy (5) 3ry-y+r- dx dy (6) dx ry-3x dy (7) 2 cos(x) = sin(x) d.x + tan () dy (8) dx 山一位

2. Classify each of equations given below as directly-integrable, separable, linear first- order, linear substitutive, homogeneous, Bernoulli, or none of them. Find the solu- tions to those equations. dy (1) x = 2y – 6x3 dy (2) 4x2-x2y? + 0. dx (3) x. dx dy Va = 3 dy (4) = y? – 2ary + a? dx dy (5) 3ry-y+r- dx dy (6) dx ry-3x dy (7) 2 cos(x) = sin(x) d.x + tan () dy (8) dx 山一位

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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Problem 6
**Classification of Differential Equations** Below are several differential equations. Your task is to classify each as directly-integrable, separable, linear first-order, linear substitutive, homogeneous, Bernoulli, or none. Additionally, find the solutions to these equations. 1. \(\frac{dy}{dx} = 2y - 6x^3\) 2. \(4x^2 - x^2y^2 + \frac{dy}{dx} = 0\) 3. \(x^2 \frac{dy}{dx} - \sqrt{x} = 3\) 4. \(\frac{dy}{dx} = y^2 - 2xy + x^2\) 5. \(3xy^3 - y + x\frac{dy}{dx} = 0\) 6. \(\frac{dy}{dx} = \frac{1}{xy - 3x}\) 7. \(2 \cos(x) \frac{dy}{dx} = \sin(x)\) 8. \(\frac{dy}{dx} = \frac{y}{x} + \tan\left(\frac{y}{x}\right)\) **Note:** Equations 3, 4, and 6 are highlighted with a black outline, indicating their importance or special consideration for solving.
Transcribed Image Text:**Classification of Differential Equations** Below are several differential equations. Your task is to classify each as directly-integrable, separable, linear first-order, linear substitutive, homogeneous, Bernoulli, or none. Additionally, find the solutions to these equations. 1. \(\frac{dy}{dx} = 2y - 6x^3\) 2. \(4x^2 - x^2y^2 + \frac{dy}{dx} = 0\) 3. \(x^2 \frac{dy}{dx} - \sqrt{x} = 3\) 4. \(\frac{dy}{dx} = y^2 - 2xy + x^2\) 5. \(3xy^3 - y + x\frac{dy}{dx} = 0\) 6. \(\frac{dy}{dx} = \frac{1}{xy - 3x}\) 7. \(2 \cos(x) \frac{dy}{dx} = \sin(x)\) 8. \(\frac{dy}{dx} = \frac{y}{x} + \tan\left(\frac{y}{x}\right)\) **Note:** Equations 3, 4, and 6 are highlighted with a black outline, indicating their importance or special consideration for solving.
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