Second order nonlinear because of (dy/dx) 2 or 1 + (dy/dx) 2 6. Second order nonlinear because of cos(r + u) 5. Third order nonlinear because of (dy/dx) 4 3. The graph of the solution defined on (−∞, ln 2) is dashed, and the graph of the solution defined on (ln 2, ∞) is solid.ġ. Thus, the solution is defined on (−∞, ln 2) or on (ln 2, ∞). Exponentiating both sides of the implicit solution we obtain 2X − 1 X − 1 = e t =⇒ 2X − 1 = Xe t − e t =⇒ (e t − 1) = (e t − 2)X =⇒ X = e t − 1 e t − 2. From y = − cos x ln(sec x + tan x) we obtain y = −1 + sin x ln(sec x + tan x) and y = tan x + cos x ln(sec x + tan x). From y = e 3x cos 2x we obtain y = 3e 3x cos 2x − 2e 3x sin 2x and y = 5e 3x cos 2x − 12e 3x sin 2x, so that y − 6y + 13y = 0. From y = e −x/2 we obtain y = − 1 2 e −x/2.
Second-order nonlinear because of ˙ x 2 11.
Second-order nonlinear because of 1/R 2 9. Second-order nonlinear because of 1 + (dy/dx) 2 8. Second-order nonlinear because of cos(r + u) 7. However, writing it in the form (v + uv − ue u)(du/dv) + u = 0, we see that it is nonlinear in u. Writing it in the form u(dv/du) + (1 + u)v = ue u we see that it is linear in v. The differential equation is first-order. However, writing it in the form (y 2 − 1)(dx/dy) + x = 0, we see that it is linear in x. Writing it in the form x(dy/dx) + y 2 = 1, we see that it is nonlinear in y because of y 2. Third-order nonlinear because of (dy/dx) 4. n-Dimensional Euclidean Space Review Exercises for Chapter 1 2.1 The Geometry of Real-Valued Functions 2.2 Limits and Continuity 2.3 Differentiation 2.4 Introduction to Paths and Curves 2.5 Properties of the Derivative 2.6 Gradients and Directional Derivatives Review Exercises for Chapter 2 3.1 Iterated Partial Derivatives 3.2 Taylor's Theorem 3.3 Extrema of Real-Valued Functions 3.4 Constrained Extrema and Lagrange Multipliers 3.5 The Implicit Function Theorem Review Exercises for Chapter 3 4.1 Acceleration and Newton's Second Law 4.2 Arc Length 4.3 Vector Fields 4.4 Divergence and Curl Review Exercises for Chapter 4 5.1 Introduction 5.2 The Double Integral Over a Rectangle 5.3 The Double Integral Over More General Regions 5.4 Changing the Order of Integration 5.5 The Triple Integral Review Exercises for Chapter 5 6.1 The Geometry of Maps from R^2 to R^2 6.2 The Change of Variables Theorem 6.3 Applications 6.4 Improper Integrals Review Exercises for Chapter 6 7.1 The Path Integral 7.2 Line Integrals 7.3 Parametrized Surfaces 7.4 Area of a Surface 7.5 Integrals of Scalar Functions Over Surfaces 7.6 Surface Integrals of Vector Fields 7.7 Applications to Differential Geometry, Physics, and Forms of Life Review Exercises for Chapter 7 8.1 Green's Theorem 8.2 Stokes' Theorem 8.3 Conservative Fields 8.4 Gauss' Theorem 8.1. 1.1 Vectors in Two- and Three-Dimensional Space 1.2 The Inner Product, Length, and Distance 1.3 Matrices, Determinants, and the Cross Product 1.4 Cylindrical and Spherical Coordinates 1.5.
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