1. Second order; linear 2. Third order; nonlinear because of (dy/dx) 4 3. Fourth order; linear 4. Second order; nonlinear because of cos(r + u) 5. Second order; nonlinear because of (dy/dx) 2 or 1 + (dy/dx) 2 6. Second order; nonlinear because of R 2 7. Third order; linear 8. Second order; nonlinear because of ˙ x 2 9. Writing the differential equation in the form x(dy/dx) + y 2 = 1, we see that it is nonlinear in y because of y 2. However, writing it in the form (y 2 − 1)(dx/dy) + x = 0, we see that it is linear in x. 10. Writing the differential equation in the form u(dv/du) + (1 + u)v = ue u we see that it is linear in v. However, writing it in the form (v + uv − ue u)(du/dv) + u = 0, we see that it is nonlinear in u. 11. From y = e −x/2 we obtain y = − 1 2 e −x/2. Then 2y + y = −e −x/2 + e −x/2 = 0. 12. From y = 6 5 − 6 5 e −20t we obtain dy/dt = 24e −20t , so that dy dt + 20y = 24e −20t + 20 6 5 − 6 5 e −20t = 24. 13. 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. 14. 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). Then y + y = tan x. 15. The domain of the function, found by solving x + 2 ≥ 0, is [−2, ∞). From y = 1 + 2(x + 2) −1/2 we have (y − x)y = (y − x)[1 + (2(x + 2) −1/2 ] = y − x + 2(y − x)(x + 2) −1/2 = y − x + 2[x + 4(x + 2) 1/2 − x](x + 2) −1/2 = y − x + 8(x + 2) 1/2 (x + 2) −1/2 = y − x + 8.
Problems 1.1.1 - 1.3.16
Problems 1.3.17 - 2.2.14
Problems 2.2.15 - 2.4.14
Problems 2.4.15 - 2.6.20
Problems 2.6.21 - 2.9.12
Problems 2.9.13 - 3.1.8
Problems 3.1.9 - 3.2.48
Problems 3.2.49 - 3.4.18
Problems 3.4.19 - 3.5.39
Problems 3.6.1 - 3.8.4
Problems 3.8.5 - 4.2.22
Problems 4.2.23 - 5.1.12
Problems 5.1.13 - 5.3.24
Problems 5.3.25 - 5.5.12
Problems 5.5.13 - 6.1.28
Problems 6.1.29 - 6.3.28
Problems 6.3.29 - 6.6.4
Problems 6.6.5 - 7.2.20
Problems 7.2.21 - 7.5.16
Problems 7.5.17 - 7.8.4
Problems 7.8.5 - 8.1.27
Problems 8.2.1 - 8.5.9
Problems 8.6.1 - 9.3.8
Problems 9.3.9 - 9.6.8
Problems 9.6.9 - 10.2.4
Problems 10.2.5 - 10.4.24
Problems 10.4.25 - 10.7.4
Problems 10.7.5 - 11.2.8
Problems 11.2.9 - 11.6.13
- Chapter 1: Introduction
- Section 1.1: Some Basic Mathematical Models; Direction Fields
- Section 1.2: Solutions of Some Differential Equations
- Section 1.3: Classification of Differential Equations
- Section 1.4: Historical Remarks
- Chapter 2: First Order Differential
Equations
- Section 2.1: Linear Equations; Method of Integrating Factors
- Section 2.2: Separable Equations
- Section 2.3: Modeling with First Order Equations
- Section 2.4: Differences Between Linear and Nonlinear Equations
- Section 2.5: Autonomous Equations and Population Dynamics
- Section 2.6: Exact Equations and Integrating Factors
- Section 2.7: Numerical Approximations: Euler's Method
- Section 2.8: The Existence and Uniqueness Theorem
- Section 2.9: First Order Difference Equations
- Chapter 3: Second Order Linear Equations
- Section 3.1: Homogeneous Equations with Constant Coefficients
- Section 3.2: Solutions of Linear Homogeneous Equations; the Wronskian
- Section 3.3: Complex Roots of the Characteristic Equation
- Section 3.4: Repeated Roots; Reduction of Order
- Section 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients
- Section 3.6: Variation of Parameters
- Section 3.7: Mechanical and Electrical Vibrations
- Section 3.8: Forced Vibrations
- Chapter 4: Higher Order Linear Equations
- Section 4.1: General Theory of nth Order Linear Equations
- Section 4.2: Homogeneous Equations with Constant Coefficients
- Section 4.3: The Method of Undetermined Coefficients
- Section 4.4: The Method of Variation of Parameters
- Chapter 5: Series Solutions of Second Order Linear Equations
- Section 5.1: Review of Power Series
- Section 5.2: Series Solutions Near an Ordinary Point, Part I
- Section 5.3: Series Solutions Near an Ordinary Point, Part II
- Section 5.4: Euler Equations; Regular Singular Points
- Section 5.5: Series Solutions Near a Regular Singular Point, Part I
- Section 5.6: Series Solutions Near a Regular Singular Point, Part II
- Section 5.7: Bessel's Equation
- Chapter 6: The Laplace Transform
- Section 6.1: Definition of the Laplace Transform
- Section 6.2: Solution of Initial Value Problems
- Section 6.3: Step Functions
- Section 6.4: Differential Equations with Discontinuous Forcing Functions
- Section 6.5: Impulse Functions
- Section 6.6: The Convolution Integral
- Chapter 7:
Systems of First Order Linear Equations
- Section 7.1: Introduction
- Section 7.2: Review of Matrices
- Section 7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors
- Section 7.4: Basic Theory of Systems of First Order Linear Equations
- Section 7.5: Homogeneous Linear Systems with Constant Coefficients
- Section 7.6: Complex Eigenvalues
- Section 7.7: Fundamental Matrices
- Section 7.8: Repeated Eigenvalues
- Section 7.9: Nonhomogeneous Linear Systems
- Chapter 8: Numerical Methods
- Section 8.1: The Euler or Tangent Line Method
- Section 8.2: Improvements on the Euler Method
- Section 8.3: The Runge-Kutta Method
- Section 8.4: Multistep Methods
- Section 8.5: Systems of First Order Equations
- Section 8.6: More on Errors; Stability
- Chapter
9: Nonlinear Differential Equations and Stability
- Section 9.1: The Phase Plane: Linear Systems
- Section 9.2: Autonomous Systems and Stability
- Section 9.3: Locally Linear Systems
- Section 9.4: Competing Species
- Section 9.5: Predator-Prey Equations
- Section 9.6: Liapunov's Second Method
- Section 9.7: Periodic Solutions and Limit Cycles
- Section 9.8: Chaos and Strange Attractors: The Lorenz Equations
- Chapter 10: Partial Differential Equations and Fourier Series
- Section 10.1: Two-Point Boundary Value Problems
- Section 10.2: Fourier Series
- Section 10.3: The Fourier Convergence Theorem
- Section 10.4: Even and Odd Functions
- Section 10.5: Separation of Variables; Heat Conduction in a Rod
- Section 10.6: Other Heat Conduction Problems
- Section 10.7: The Wave Equation: Vibrations of an Elastic String
- Section 10.8: Laplace's Equation
- Chapter 11: Boundary Value Problems and Sturm-Liouville Theory
- Section 11.1: The Occurence of Two-Point Boundary Value Problems
- Section 11.2: Sturm-Liouville Boundary Value Problems
- Section 11.3: Nonhomogeneous Boundary Value Problems
- Section 11.4: Singular Sturm-Liouville Problems
- Section 11.5: Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
- Section 11.6: Series of Orthogonal Functions: Mean Convergence
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