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## Contents

0.2.4 Exercises

0.3.1 Exercises

1.1.1 Exercises
1.2 Slope ﬁelds
1.2.1 Slope ﬁelds

1.2.3 Exercises

1.3.1 Separable equations
1.3.2 Implicit solutions
1.3.3 Examples
1.3.4 Exercises

1.4.1 Exercises
1.5 Substitution
1.5.1 Substitution
1.5.2 Bernoulli equations

1.5.4 Exercises

1.6.1 Exercises

1.7.1 Exercises

1.8.2 Integrating factors
1.8.3 Exercises

2.1.1 Exercises

2.2.2 Complex roots
2.2.3 Exercises

2.3.1 Linear independence

2.3.3 Exercises

2.4.1 Some examples
2.4.2 Free undamped motion
2.4.3 Free damped motion
2.4.4 Exercises

2.5.4 Exercises

2.6.3 Exercises

3.1.1 Exercises

3.2.1 Matrices and vectors

3.2.3 The determinant

3.2.6 Exercises

3.3.1 Exercises

3.4.3 Complex eigenvalues
3.4.4 Exercises

3.5.1 Exercises

3.6.2 Examples
3.6.3 Forced oscillations
3.6.4 Exercises

3.7.3 Exercises

3.8.1 Deﬁnition
3.8.2 Simple cases
3.8.3 General matrices

3.8.5 Approximations
3.8.6 Exercises

3.9.4 Exercises

4.1.2 Eigenvalue problems

4.1.4 Fredholm alternative
4.1.5 Application
4.1.6 Exercises

4.2.4 Exercises

4.3.1 -periodic functions
4.3.2 Convergence

4.3.5 Exercises

4.4.3 Application
4.4.4 Exercises

4.5.2 Resonance
4.5.3 Exercises

4.6.3 Insulated ends
4.6.4 Exercises

4.7.1 Exercises

4.8.1 Change of variables
4.8.2 D’Alembert’s formula

4.8.4 Exercises

4.9.1 Exercises

4.10.2 Series solution
4.10.3 Poisson kernel
4.10.4 Exercises

5.1.2 Orthogonality
5.1.3 Fredholm alternative
5.1.4 Eigenfunction series
5.1.5 Exercises

5.2.1 Exercises

5.3.3 Exercises

6.1.1 The transform

6.1.4 Exercises

6.2.4 Transfer functions

6.2.6 Exercises
6.3 Convolution
6.3.1 The convolution
6.3.2 Solving ODEs

6.3.4 Exercises

6.4.1 Rectangular pulse
6.4.2 The delta function
6.4.3 Impulse response

6.4.5 Exercises

7.1 Power series
7.1.1 Deﬁnition

7.1.3 Analytic functions

7.1.6 Exercises

7.2.1 Exercises

7.3.1 Examples

7.3.3 Bessel functions
7.3.4 Exercises

8.1.2 Linearization
8.1.3 Exercises

8.2.5 Exercises

8.3.1 Pendulum

8.3.3 Exercises
8.4 Limit cycles
8.4.1 Exercises
8.5 Chaos

8.5.2 The Lorenz system
8.5.3 Exercises