Chapter 0.
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Preliminaries |
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1. |
Introduction |
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2 |
Complex numbers |
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3 |
Functions |
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4 |
Polynomials |
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5. |
Complex series and the exponential function |
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6. |
Determinants |
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7. |
Remarks on methods of discovery and proof |
Chapter 1. |
Introduction--Linear Equations of the First Order |
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1. |
Introduction |
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2. |
Differential equations |
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3. |
Problems associated with differential equations |
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4. |
Linear equations of the first order |
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5. |
The equation y'+ay=0 |
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6. |
The equation y'+ay=b(x) |
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7. |
The general linear equation of the first order |
Chapter 2. |
Linear Equations with Constant Coefficients |
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1. |
Introduction |
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2. |
The second order homogeneous equation |
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3. |
Initial value problems for second order equations |
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4. |
Linear dependence and independence |
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5. |
A formula for the Wronskian |
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6. |
The non-homogeneous equation of order two |
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7. |
The homogeneous equation of order n |
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8. |
Initial value problems for n-th order equations |
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9. |
Equations with real constants |
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10. |
The non-homogeneous equation of order n |
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11. |
A special method for solving the non-homogeneous equation |
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12. |
Algebra of constant coefficient operators |
Chapter 3. |
Linear Equations with Variable Coefficients |
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1. |
Introduction |
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2. |
Initial value problems for the homogeneous equation |
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3. |
Solutions of the homogeneous equation |
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4. |
The Wronskian and linear independence |
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5. |
Reduction of the order of a homogeneous equation |
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6. |
The non-homogeneous equation |
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7. |
Homogeneous equations with analytic coefficients |
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8. |
The Legendre equation |
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9. |
Justification of the power series method |
Chapter 4. |
Linear Equations with Regular Singular Points |
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1. |
Introduction |
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2. |
The Euler equation |
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3. |
Second order equations with regular singular points--an example |
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4. |
Second order equations with regular singular points--the general case |
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5. |
A convergence proof |
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6. |
The exceptional cases |
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7. |
The Bessel equation |
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8. |
The Bessel equation (continued) |
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9. |
Regular singular points at infinity |
Chapter 5. |
Existence and Uniqueness of Solutions to First Order Equations |
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1. |
Introduction |
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2. |
Equations with variables separated |
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3. |
Exact equations |
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4. |
The method of successive approximations |
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5. |
The Lipschitz condition |
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6. |
Convergence of the successive approximations |
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7. |
Non-local existence of solutions |
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8. |
Approximations to, and uniqueness of, solutions |
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9. |
Equations with complex-valued functions |
Chapter 6. |
Existence and Uniqueness of Solutions to Systems and n-th Order Equations |
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1. |
Introduction |
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2. |
An example--central forces and planetary motion |
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3. |
Some special equations |
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4. |
Complex n-dimensional space |
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5. |
Systems as vector equations |
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6. |
Existence and uniqueness of solutions to systems |
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7. |
Existence and uniqueness for linear systems |
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8. |
Equations of order n |
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References; Answers to Exercises; Index |
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