# Differential Equations

## First order DEs

### IVPs, Direction Fields, Isoclines

### Direction Fields, Autonomous DEs

### Separable DEs, Exact DEs, Linear 1st order DEs

### Numerical method: Euler (or Constant Slope)

### Applications (Growth/Cooling/Circuits/Falling body)

## Higher order DEs

### IVPs/General solutions, Basic theory

### Numerical methods for higher order DEs

### Constant coefficient case: Undetermined Coefficients

### Application: springs (free, damped, forced, pure resonance)

### Application: Electrical Circuits

## Laplace Transform (LT) methods

### Inverse Laplace & Derivatives

### 1st Translation Thrm

### Partial Fractions, completing the square

### Unit Step Functions

SAGE can define piecewise functions like $$

- \begin{cases}
x \ {\mapsto}\ \sin \left( \frac{2} \right) &\text{on (0, 1)}\cr x \ {\mapsto}\ 1 - {\left( x - 1 \right)}^{2} &\text{on (1, 3)}\cr x \ {\mapsto}\ -x &\text{on (3, 5)} \end{cases}

$$ as follows:

sage: f(x) = sin(x*pi/2) sage: g(x) = 1-(x-1)^2 sage: h(x) = -x sage: P = Piecewise([[(0,1), f], [(1,3),g], [(3,5), h]]) sage: latex(P)

### 2nd Translation Theorem

### Derivative thrms, Solving DEs

### Convolution theorem

### Dirac Delta Function

### Application: Lanchester's equations

### Application: Electrical networks

## PDEs

### Separation of Variables

### Heat Equation., Fourier's solution

### Fourier Series

### Convergence, Dirichlet's theorem

### Fourier Sine Series, Fourier Cosine Series

### Heat Eqn. Ends at Zero