Three pillars: Qualitative, Analytic, Numerical.
I. General tools
1. Complex numbers: z = a + bi; e^{i theta} = cos(theta) + i sin(theta).
2. Polar identity: a cos(t) + b sin(t) = A cos(t - phi), where A is the hypotenuse of the right triangle with legs a, b and phi is the angle opposite b.
3. Fourier series: A function of period 2 pi has an expression (a0)/2 + a1 cos(t) + b1 sin(t) + a2 cos(2t) + b2 sin(2t) + .... an = (1/pi) integral_{-pi}^{pi} cos(t) f(t) dt ; bn with sin.
4. Laplace transform, delta, convolution: see III 3. below.
II. Linear equations
1. Standard form Ly = b(t) where L is the linear differential operator D^n + a{n-1}(t)D^{n-1} + ... + a0(t)I. L represents the system, b(t) an outside influence. Solutions exist. n independent ones, y1, ... yn, in the homogeneous case b = 0. General solution is yp + linear comb. of yk's. Second order model: spring/dashpot/mass.
2. Constant coefficient case: controlled by the characteristic polynomial p(lambda) = lambda^n + ... + a0; L = p(D).
a. Homogeneous case: normal modes, e^{rt} where r is root of p. If r = a+bi, r-bar is a root also, and real solutions are e^{at}cos(bt), e^{at}sin(bt). Real part positive: exponential growth; negative: decay. Imaginary part nonzero: oscillation. Repeated roots: multiply by t.
b. Inhomogeneous case: Undetermined coefficients b(t) poly of deg m and a0 not zero: there's exactly one sol which is poly of deg m. b(t) = e^{st} and p(s) not zero: y = e^{st}/p(s). ESL: p(D)(e^{st}u) = e^{st}p(D+sI)u.
3. Laplace transform methodology.
a. L(f'(t);s) = sL(f(t);s) - f(0).
b. p(D)x = b(t), rest initial conditions: X(s) = W(s)B(s), W(s) = 1/p(s) the transfer function.
c. Convolution; L(f*g) = L(f)L(g); x(t) = w(t)*b(t); decay model.
d. Delta; interpretation as lump sum or impulse; delta = u'; integral delta_a(t) g(t) dt = g(a); delta_a * f = f_a.
e. Methods of finding inverse Laplace transform: partial fractions, convolution.
4. First order nonconstant coefficient inhomogeneous: y' + p(t)y = q(t). Bank example. Integrating factors.
5. 2-d systems: u-dot = Au, A a 2x2 matrix. The characteristic polynomial is p(lambda) = lambda^2 - (tr A)lambda + (det A). Behavior controlled by eigenvalues; eigenvectors. Phase portraits: spiral, center, proper node, improper node, star node, saddle; degenerate cases. Stability.
6. IVP's: Fundamental matrices, e^{At}.
7. PDE's: Heat equation. Boundary conditions; separation of variables.
III. Nonlinear equations
1. Separable equations.
2. Higher order reducible equations: eg ma = GMm/x^2: v = dx/dt, then a = dv/dt = v(dv/dx) leads to first order equation.
3. Systems: critical point analysis of phase portrait; Jacobian.
IV. Numerical methods: Euler, Heun, Runge-Kutta.