Lecture 1: Course introduction. Outline. Classification of differential equations: linear/nonlinear, first order etc. Separable equations. Modeling with differential equations.

Lecture 2: Geometric methods. Direction fields, integral curves. Integral curves do not cross or touch. Examples. Computer demo: isoclines.
First order linear equations. Method of integrating factors.

Lecture 3: More about first order linear order equaitons. Exact equations. Exact equations and integrating factors.

Lecture 4: Autonomous equations. Criticial points. Phase portrait. Classification of critical points: asymptotically stable, semistable, unstable. Logistic growth. Computer demo: phase lines.

Lecture 5: Second order linear homogeneous differential equations and IVP. Superposition of solutions. Wronskian is given by a 2 x 2 determinant. Fundamental pairs of solutions have non-zero Wronskian.
Second order linear homogeneous differential equations with constant coefficients. Models spring-mass system. Characteristic equation. General solution in terms of solutions to the characteristic equation (3 cases: real distinct roots, repeated roots, complex roots).

Lecture 6: Case of complex roots. Complex numbers, complex exponentials. Real valued solutions are found by taking real and imaginary part of the complex valued solutions. Oscillations. Review for midterm.

Lecture 7: Midterm 1.

Lecture 8: General theory of inhomogeneous equations. Solutions are of the form y=y_p+y_h. Finding the particular solution y_p by undertmined coefficients. The exponential, polynomial, trigonometric and mixed cases. See Table 3.5.1 on page 182.

Lecture 9: Variation of parameters and examples. I showed how to look for solutions of the form y=u_1 y_1+u_2 y_2, where y_1, y_2 solve the homogeneous equation and u_1, u_2 are functions to be determined. I determined u_1, u_2.
Matrices and vectors. Addition, scalar multiples. More about 2x2 matrices: products, determinants.

Lecture 10: Invertible matrices. Calculating inverses. Solving systems of linear equations. Eigenvalues. Eigenvectors. Characteristic polynomials.
Systems of first order linear equations. Modeling. A first order linear system becomes a second order differetial equation and conversely.

Lecture 11: Solutions of 2-dimensional systems of first order ODEs are parametric curves. Solving homogeneous linear first order systems: the case of distinct real eigenvalues and the case of complex eigenvalues.

Lecture 12: Solving homogeneous linear first order systems: the case of repetead eigenvalues. How to sketch solutions: stable and unstable sources/sinks for distinct real eigenvalues with the same sign, saddles for real eigenvalues of opposite signs, spirals for complex eigenvalues with nonzero real part. Computer demo: phase portraits.

Lecture 13: How to solve and sketch phase portrait of 2-dimensional systems.

Lecture 14: Review for midterm2: homogeneous linear second order equations (superpostion, repeated roots); unhomogeneous linear second order equations via undetermined coefficients and via variation of parameters; operations with matrices and vectors, eigenvalues, eigenvectors, linear independence; 2-dimensional linear systems (superposition, how to solve, phase portraits).

Lecture 15: Midterm 2.

Lecture 16: Inhomogeneous systems. Undetermined coefficients. Series methods. Examples.

Lecture 17: More about Taylor series and ODEs. Radius of convergence. Solving differential equations using series: finding recursive formulas between coefficients. Table.

Lecture 18: Laplace transform. Functions of exponential growth. I calculated the Laplace transform of 1, e^{at}, cos (at), sin (at) and t^n. hift rule: Laplace of e^{at} f(t) is F(s-a). Laplace transforms of derivatives. Inverse Laplace transform and partial fractions. Using Laplace transform to solve initial value problems.

Lecture 19: Step functions. Laplace transform of step functions. The inverse Laplace transform of e^{-sa}F(s) is u_a(t)f(t-a). How to solve differential equations involving step functions.

Lecture 20: Review.