Lecture 1: ODE Review

• ODE class: recipe-like methods
  • classify, guess, verify
  • limited scope
• Asymptotic methods: approximate

Differential Equations

1. Separable equations $$ \begin{aligned} y'(x) &= A(x)B(y) \ \int \frac{1}{B(y)} \mathrm{d}y &= \int A(x) \mathrm{d}x \ \implies F(y) &= G(x) + c_0 \end{aligned} $$

   • general solution: \(y = y(x; c_0)\)
   • Initial Value Problem (IVP)

   • \(n\)-th order ODE $$ y^{(n)} = F(x, y, y', \ldots, y^{(n-1)}) $$

   • linear $$ y^{(n)} + p_{n-1}(x) y^{(n-1)} + \ldots + p_1(x) y' + p_0(x) y = f(x) $$

     • homogeneous: \(f(x) = 0\)

Linear Homogeneous ODEs

Wronskians

\[ W(x) = W(y_1, y_2, \ldots, y_n) = \begin{vmatrix} y_1 & y_2 & \ldots & y_n \\ y_1' & y_2' & \ldots & y_n' \\ \vdots & \vdots & \ddots & \vdots \\ y_1^{(n-1)} & y_2^{(n-1)} & \ldots & y_n^{(n-1)} \end{vmatrix} \]

• \(\{y_1, y_2, \ldots, y_n\}\) are linearly dependent \(\iff W(x) = 0\) for all \(x\).
• \(\{y_1, y_2, \ldots, y_n\}\) are linearly independent \(\iff W(x) \neq 0\) except at isolated points.

Abel's Formula

\[ W'(x) = -p_{n-1}(x) W(x) \implies W(x) = W(x_0) \exp\left(-\int_{x_0}^x p_{n-1}(t) \mathrm{d}t\right) \]

the general solution will remain to be general solution for all \(x\) s.t., \(p_{n-1}(x)\) is non-singular.

IVP is ill-posed if \(W(x_0) = 0\).

Example

\[ y'' - \frac{1+x}{x} y' + \frac{1}{x} y = 0 \]

\(p_1(x) = -\frac{1+x}{x}\) is singular at \(x=0\).

Abel's formula gives \(W(x) = Cx e^x\). At \(x_0 = 0\), \(W(0) = 0\).

Check:

• \(y(0) = 1, y'(0) = 2\) has no solution.
• \(y(0) = 1, y'(0) = 1\) has more than one solution.

Boundary value problem (BVP) is global.

• \(y_k(x) = e^{r_k x}\) ansatz for constant coefficient ODEs, where \(r_k\) are roots of characteristic polynomial.
• Euler equations
  • e.g. $ y'' + \frac{y}{4x^2} = 0$
  • is invariant w.r.t. the scaling \(x \mapsto ax\).
  • \(y = x^r\) or \(y = x^r (\ln x)^s\) ansatz.
• Reduction of order
  • given one solution \(y_1(x)\), try \(y = v(x) y_1(x)\).

Inhomogeneous ODEs

\[ y(x) = \sum_{j=1}^n c_j y_j(x) + y_{\text{p}}(x) \]

Finding \(y_{\text{p}}(x)\):

1. Variation of parameters $$ y_{\text{p}}(x) = \sum_{j=1}^n c_j(x) y_j(x) $$
2. Green's function for \(Ly = f\), \(G(x, a)\) at \(a\) is defined as the solution to $$ L G(x, a) = \delta(x - a) $$

Remarks on delta function

\(\delta(x - a)\) is the Dirac delta function, "derivative"¹ of the Heaviside step function \(H(x - a)\).

Note that the jump discontinuity of Heaviside function is milder than the \(\infty\)-singularity of \(\delta\) function.

• ramp function:

\[ \int_{-\infty}^x h(t-a) \, \mathrm{d}t = r(x - a) = \begin{cases} 0, & x < a \\ x - a, & x \geq a \end{cases} \]

The following hierarchy shows the increasing smoothness (decreasing singularity):

\[ \delta(x - a) \to H(x - a) \to r(x - a) \to C^k \to C^\infty \]

integration is a smoothing operation.

Important property

\[ \int_{-\infty}^{\infty} \delta(x - a) f(x) \, \mathrm{d}x = f(a) \]

This is the weak definition of \(\delta\) function.

distributions are functionals on \(C_0^\infty(\mathbb{R})\)

\[ \mathcal{D}'(\mathbb{R}) = \{\text{linear functionals on } C_0^\infty(\mathbb{R})\} \]

Dirac delta function can be expressed as a limit of smoother functions, e.g.

\[ \delta(x - a) = \lim_{\epsilon \to 0} \frac{1}{\sqrt{\pi \epsilon}} e^{-\frac{(x - a)^2}{\epsilon}} \]

Set \(y(x) = \int_a^b G(x, a) f(a) \, \mathrm{d}a\).

Then $$ \begin{aligned} L y(x) &= L \int_a^b G(x, a) f(a) \, \mathrm{d}a \ &= \int_a^b L G(x, a) f(a) \, \mathrm{d}a \ &= \int_a^b \delta(x - a) f(a) \, \mathrm{d}a = f(x) \end{aligned} $$

Solve \(LG = \delta(x - a)\)

e.g. \(\(L = \frac{\mathrm{d}^2}{\mathrm{d}x^2} + p_1(x) \frac{\mathrm{d}}{\mathrm{d}x} + p_0(x)\)\)

\[ G(x, a) = \begin{cases} A_1 y_1(x) + A_2 y_2(x), & x < a \\ B_1 y_1(x) + B_2 y_2(x), & x > a \end{cases} \]

\[ G'' + p_1(x) G' + p_0(x) G = \delta(x - a) \]

\(G''\) should have a singularity similar to \(\delta(x - a)\), that is,

\[ G'' \sim \delta(x - a). \]

Likewise, \(G'\) should have a jump discontinuity like \(H(x - a)\), and \(G\) should be continuous like \(r(x - a)\).

Thus,

\[ A_1 y_1(a) + A_2 y_2(a) = B_1 y_1(a) + B_2 y_2(a) \]

Bernoulli Eq.

\[y' + p(x) y = q(x) y^n\]

substitution: \(v = y^{1-n}\)

Riccati Eq.

\[y' = a(x) y^2 + b(x) y + c(x)\]

substitute: \(\(y = - \frac{W'(x)}{a(x) W(x)}\)\)

\[\implies W'' - \left(\frac{a'(x)}{a(x)} + b(x)\right) W' + a(x) c(x) W = 0\]

Summary

1. For linear equations (\(n\)-th order), general sol. with \(n\) constants.
   • Interval of well-posedness of the sol. extends from a local neighborhood of \(x_0\) to the maximal interval where \(p_k(x)\) are continuous.
2. nonlinear equations, general sol. with \(n\) constants + other sol.
   • sol. could develop autonomous singularities.

---

1. Weak derivative in the distribution theory. ↩