Topics covered in 18.03 May 11, 1999 Edward Early

1. Separable equations, e.g. y′ = xy.

2. Direction fields, isoclines, e.g. y′ = \frac{x}{y}.

3. Linear first order equations and integrating factors, e.g. y′ + xy = x.

4. Substitution: homogeneous, use v = \frac{y}{x}, e.g. y′ = \frac{x^{2}+y^{2}}{x^{2}−y^{2}} .

5. Autonomous equations and the phase line, e.g. y′ = y^{2} − y.

6. Numerical methods: Euler, Improved Euler (Heun), Runge-Kutta. e.g. y′ = x.

7. Second order linear equations, homogeneous, e.g. y′′ + 5y′ + 6y = 0.

8. Complex numbers, complex exponentials, Euler’s formula (e^{ix} = cos x + i sin x).

9. Non-real and repeated roots, e.g. y′′ + y = 0 and y′′ + 2y′ + y = 0.

10. Initial value problems, e.g. y′′ + y = 0, y(0) = 0, y′(0) = 1.

11. Undetermined coefficients, e.g. y′′ + 2y′ + y = e^{x}. First solve associated homogeneous equation, take a guess at form of particular solution (ansatz), plug in to find particular solution, then combine for general solution. Multiply guess by x if necessary so that it isn’t part of the homogeneous solution.

12. Operators, exponential shift law (ESL): D(e^{ax}y) = e^{ax}(D + aI)(y), e.g. y′′ + 2y′ + y = e^{x}. Bonus: Fibonacci numbers: F_{0} = 0, F_{1} = 1, F_{n} = F_{n−1}+F_{n−2} for n > 1; F_{n} = \frac{1}{\sqrt{5}} ((\frac{1+\sqrt{5}}{2})^{n} − (\frac{1−\sqrt{5}}{2})^{n}).

13. Linear equations with nonconstant coefficients, e.g. Airy equation y′′ + xy = 0. 13\frac{1}{2}. Non-linear equations, substitute v = y′ or something clever.

14. Systems of first order ODEs, e.g. x′ = x + y, y′ = x + 2y.

15. Eigenvalues (roots of characteristic polynomial) and eigenvectors (Av = λv).

16. Complex or repeated eigenvalues, e.g. [ \begin{matrix}2 & −1 \\ 1 & 0 \end{matrix} ], [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] .

17. Phase portraits. See class handout for more information. Nodes: proper, star (complete), improper (incomplete). Saddle, center, spiral, degenerate.

18. Nonlinear systems, critical points, linearization by translation or Jacobian, e.g. x′ = x + xy, y′ = y − xy.

19. Matrix exponential e^{At}, fundamental matrix Φ(t), initial value problems.

20. Laplace transform, L(f) = ∫_{0}^{∞} e^{−st}f(t)dt. e.g. y′′ + 2y′ + y = e^{x}. Properties: Linear, preserves Re and Im, s shift, t shift, s derivative, t derivative.

21. Discontinuous and generalized functions, e.g. δ, u_{a}. L(δ) = 1.

22. Convolution, (f ∗ g) = ∫_{0}^{t} f(τ)g(τ − t)dτ. L(f ∗ g) = L(f) · L(g), (f ∗ δ_{a}) = f(a).

23. Transfer function W (s), weight function w(t), and Duhamel’s principle x(t) = w(t) ∗ f(t).

24. Fourier series, f(x) = \frac{a_{0}}{2} + ∑_{n=1}^{∞}(a_{n} cos(nt) + b_{n} sin(nt)), where f(x) is periodic with period 2π, e.g. square wave. ∫_{−π}^{π} f(t)dt = πa_{0}, ∫_{−π}^{π} cos(mt)f(t)dt = πa_{m}, ∫_{−π}^{π} sin(mt)f(t)dt = πb_{m} for m > 0. Bonus: Formulas for π: \frac{π}{4} = 1 − \frac{1}{3} + \frac{1}{5} − + ⋯, \frac{π^{2}}{6} = 1 + \frac{1}{4} + \frac{1}{9} + · · ·.

25. Partial differential equations (PDEs). Heat equation, wave equation.