Still Non-Linear ODE

How would we solve equations like:

\begin{equation} \begin{cases} y’’ - 2xy’ + 2\lambda y = 0 \\ y’’ - xy = 0 \end{cases} \end{equation}

Taylor Series

Its time to have a blast from the past! Taylor Series time.

\begin{equation} p_{n}(x) = \sum_{i=0}^{n} \frac{f^{(n)}(0) x^{n}}{n!} \end{equation}

Taylor’s Theorem with Remainder gives us that, at some \(n\), \(|f(x) - p_{n}(x)|\) is bounded.

\begin{equation} |x(t+h) - (x(t) + h x’(t))| \leq Ch \end{equation}

Insight: if your derivatives are bounded, then at high values of \(j\) we have \(\frac{f^{(j)}\qty(0)}{n!}\) tends eventually towards zero as \(n\) increases.

A Partial Differential Equation is a Differential Equation which has more than one independent variable: $u(x,y), u(t,x,y), …$

For instance:

\begin{equation} \pdv{U}{t} = \alpha \pdv[2]{U}{x} \end{equation}

Key Intuition

• PDEs may have no solutions (unlike Uniqueness and Existance for ODEs)
• yet, usually, there are too many solutions—so… how do you describe all solutions?
• usually, there are no explicit formulas

Laplacian of \(u(x,y)\)

Laplacian of \(u(x,y)\)

Examples

Heat Equation

See Heat Equation

Wave Equation

see Wave Equation

Transport Equation

\begin{equation} \pdv{u}{t} = \pdv{u}{x} \end{equation}