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Let y(x) = cx³ be a one-parameter family of cubic polynomials. (a) Show that the family of curves that is orthogonal at every point of the (x, y)-plane to this family of cubics satisfies the following differential equation. y' = - X 3y The slope y'(x) = m = y . Using c = gives m = x3 of functions must satisfy the differential equation y' X 3y . Therefore, the orthogonal family

Let y(x) = cx³ be a one-parameter family of cubic polynomials. (a) Show that the family of curves that is orthogonal at every point of the (x, y)-plane to this family of cubics satisfies the following differential equation. y' = - X 3y The slope y'(x) = m = y . Using c = gives m = x3 of functions must satisfy the differential equation y' X 3y . Therefore, the orthogonal family

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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Let y(x) = cx³ be a one-parameter family of cubic polynomials. (a) Show that the family of curves that is orthogonal at every point of the (x, y)-plane to this family of cubics satisfies the following differential equation. X 3y The slope y'(x) = m = of functions must satisfy the differential equation y' y . Using c = gives m = x3 Using implicit differentiation on x² + 3y² = C gives equation for y' gives y' = = (b) Show that the set of curves x² + 3y² = C is a one-parameter family satisfying the differential equation y' part (a). X 1 Зу' V X 3y as desired. + (c) Either using a curve-plotting program, or by hand, sketch the curves y = x² + 3y² = C for C = 1, 2, 3. Therefore, the orthogonal family cx³ for c = X 3y V y' = 0. Solving this ±1, c = ±2 and the curves in
Transcribed Image Text:Let y(x) = cx³ be a one-parameter family of cubic polynomials. (a) Show that the family of curves that is orthogonal at every point of the (x, y)-plane to this family of cubics satisfies the following differential equation. X 3y The slope y'(x) = m = of functions must satisfy the differential equation y' y . Using c = gives m = x3 Using implicit differentiation on x² + 3y² = C gives equation for y' gives y' = = (b) Show that the set of curves x² + 3y² = C is a one-parameter family satisfying the differential equation y' part (a). X 1 Зу' V X 3y as desired. + (c) Either using a curve-plotting program, or by hand, sketch the curves y = x² + 3y² = C for C = 1, 2, 3. Therefore, the orthogonal family cx³ for c = X 3y V y' = 0. Solving this ±1, c = ±2 and the curves in
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