18.03 Study Outline for Second Hour Exam

1. Ideas of linearity

a. General form of a linear ODE:

Ly = y^{(n)} + p_{1}(x)y^{(n−1)} + ⋯ + p_{n}(x)y = r(x).  (1)

b. p_{1}(x), … p_{n}(x) are the “coefficients.” Homogeneous means r(x) = 0:

y^{(n)} + p_{1}(x)y^{(n−1)} + ⋯ + p_{n}(x)y = 0.  (2)

c. Solutions y_{p} exist whenever the coefficients are reasonable. “Superposition”: If y_{1} solves Ly = r_{1}(x), y_{2} solves Ly = r_{2}(x), and c is a constant, then y_{1} + y_{2} solves Ly = r_{1}(x) + r_{2}(x) and cy_{1} solves Ly = cr_{1}(x).

d. There are n linearly independent solutions y_{1}, …, y_{n} to (2). {y_{1}, …, y_{n}} is linearly independent if the only way the linear combination a_{1}y_{1} + ⋯ + a_{n}y_{n} can be 0 is for a_{1} = ⋯ = a_{n} = 0.

e. “Superposition”: The general solution to (1) is y_{p} + a_{1}y_{1} + ⋯ + a_{n}y_{n}, for any choice of y_{p}, y_{1}, …, y_{n} as above.

2. Complex numbers

a. (1, 0) in the plane is called 1, (0, 1) is called i, so (a, b) is called a + bi. This gives addition. Multiplication is determined by i^{2} = −1 and the standard rules. Re (a + bi) = a, Im (a + bi) = b (where a, b are understood to be real).

b. Complex conjugation \overline{a + bi} = a − bi (a, b real). \overline{z + w} = z̄ + w̄, \overline{zw} = z̄w̄. The magnitude or modulus or absolute value |z| of z is the distance to 0, and zz̄ = |z|^{2}. Re z = (z + z̄)/2, Im z = (z − z̄)/2i. Divide by multiplying numerator and denominator by the conjugate of the denominator.

c. Polar rule for multiplication: Magnitudes multiply, angles add.

d. Compute derivatives of complex valued functions coordinatewise. For fixed complex r, e^{rt} is the solution to the IVP ż = rz, z(0) = 1. e^{(a+bi)t} = e^{at}(cos(bt) + i sin(bt)). e^{z+w} = e^{z}e^{2}, e^{r̄t} = \overline{e^{rt}}.

3. Constant coefficient homogeneous equations

a. e^{rx} is a solution to (2) exactly when r is a root of the characteristic polynomial

r^{n} + p_{1}r^{n−1} + ⋯ + p_{n−1}r + p_{n}.  (3)

These are linearly independent “normal modes.”

b. If r = a + bi is a root and i ≠ 0, then r̄ = a − bi is a root too and so the real and imaginary parts of e^{rx} are linearly independent solutions: e^{ax} cos(bx), e^{ax} sin(bx).

c. If r has multiplicity k as a root of (3), then e^{rx}, xe^{rx}, …, x^{k−1}e^{rx}, are independent solutions to (2).

d. If Re r < 0 the normal mode is damped: it falls exponentially to zero as x → ∞. If Re r = 0 it is periodic. If Re r > 0 it blows up. If Im r ≠ 0 it is oscillatory. Damped oscillation is “underdamping”; if r is real and negative we have “overdamping.”

4. Initial value problems: second order

a. The Wronskian of two functions y_{1}, y_{2} is w(x) = y′_{1}y_{2} − y_{1}y′_{2}. If y_{1}, y_{2} are solutions to y′′ + p(x)y′ + q(x) = 0, then w is either never zero (and {y_{1}, y_{2}} is linearly independent) or always zero (and {y_{1}, y_{2}} is linearly dependent). Abel’s equation: w′ + p(x)w = 0: so if p = 0 then w is constant.

b. Given any x_{0} in the range of definition of the equation and any a, b, there is a unique solution y to (1) such that y(x_{0}) = a, y′(x_{0}) = b. One finds it by solving for a and b in y = y_{p} + ay_{1} + by_{2}.

c. y_{1}, y_{2} is normalized at x_{0} if y_{1}(x_{0}) = 1, y′_{1}(x_{0}) = 0, y_{2}(x_{0}) = 0, y′_{2}(x_{0}) = 1. Then y = ay_{1} + by_{2} has y(x_{0}) = a, y′(x_{0}) = b.

d. For p > 0 solutions to y′′ + py′ + qy = e^{iω_{0}x} have “steady state” and “transient” components. Steady state exhibits an amplitude multiplier (“gain”) and a phase lag. Practical resonance means ω_{0} is tuned to maximize gain. Beats, pure resonance may occur if p = 0.

5. Operators

a. Examples: Dy = y′; Iy = y (the “identity operator”); M_{f}y = f(x)y, for a fixed function f(x). Operators compose, so D^{2}y = y′′, and add, so (D^{2}+2D+3I)y = y′′+2y′+3y. An operator F is linear if F (u + v) = F u + F v and F (cu) = cF u (c constant). D, I, M_{f} are linear; Sy = y^{2}, Ay = y + 1 are not. The general CCLDO is L = D^{n} + p_{1}D^{n−1} + ⋯ + p_{n}I, where p_{1}, …, p_{n} are constants. Its characteristic polynomial is f(r) = r^{n} + p_{1}r^{n−1} + ⋯ + p_{n}, and L = f(D).

b. De^{rx} = re^{rx}. If L is a CCLDO with characteristic polynomial f, then Le^{rx} = f(r)e^{rx}. More generally, the Exponential Shift Law says f(D)(e^{rx}u) = e^{rx}f(D + rI)u.

6. Undetermined coefficients

a. If L is a CCLDO with p_{n} ≠ 0 and r(x) is a polynomial of degree m, then Ly = r(x) has exactly one solution which is polynomial of degree m.

b. If 0 is a root of f of multiplicity k, then regard Ly = r(x) as an equation of order n − k for y^{(k)}, solve it, and integrate k times. (Ly = r(x) is “reducible.”)

c. If s is not a root of f, then Ly = e^{sx} has solution e^{sx}/f(s).

d. The ESL shows that Ly = e^{sx}q(x) is equivalent to f(D − sI)y = q(x). This is useful if s is a root of f (when it leads to a reducible equation) or if s ≠ 0 and q is not constant (when it eliminates the exponential; if q is polynomial we are in case b. or c.).

7. Nonconstant coefficients, nonlinear equations

a. No normal modes, no characteristic polynomial; superposition still holds if linear.

b. Homogeneous Euler-Cauchy: x^{2}y′′ + pxy′ + qy = 0: y = x^{r} is a solution exactly when r(r − 1) + pr + q = 0. Use x = e^{ln x} (for x > 0) to give meaning to this, even if r is complex. Repeated roots can be handled by

c. Reduction of order: If y_{1} is one solution of Ly = 0, try to find another by writing y = y_{1}u and deriving an ODE for u.

d. A second order equation (even nonlinear) can be reduced to a pair of first order equations if it is reducible: missing y (when it is really a first order equation for v = y′) or y′ (when it is a first order equation for y′ as a function of y).