# Documentation

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## DDE of Neutral Type

This example shows how to use `ddensd` to solve the neutral DDE presented by Paul [1] for 0 ≤ tπ.

Click `ddex4.m` or type `edit ddex4.m` in a command window to view the code for this example in an editor.

The equation is

y '(t) = 1 + y(t) – 2y(t/2)2y '(tπ)

with history:

y(t) = cos (t) for t ≤ 0.

1. Create a new program file in the editor. This file will contain a main function and four local functions.

2. Define the first-order DDE as a local function.

```function yp = ddefun(t,y,ydel,ypdel) yp = 1 + y - 2*ydel^2 - ypdel; end```
3. Define the solution delay as a local function.

```function dy = dely(t,y) dy = t/2; end```
4. Define the derivative delay as a local function.

```function dyp = delyp(t,y) dyp = t-pi; end```
5. Define the solution history as a local function.

```function y = history(t) y = cos(t); end```
6. Define the interval of integration and solve the DDE using the `ddensd` function. Add this code to the main function.

```tspan = [0 pi]; sol = ddensd(@ddefun,@dely,@delyp,@history,tspan);```
7. Evaluate the solution at 100 equally spaced points between 0 and π. Add this code to the main function.

```tn = linspace(0,pi); yn = deval(sol,tn);```
8. Plot the results. Add this code to the main function.

```figure plot(tn,yn); xlim([0 pi]); ylim([-1.2 1.2]) xlabel('time t'); ylabel('solution y'); title('Example of Paul with 1 equation and 2 delay functions')```
9. Run your program to calculate the solution and display the plot.

## References

[1] Paul, C.A.H. “A Test Set of Functional Differential Equations.” Numerical Analysis Reports. No. 243. Manchester, UK: Math Department, University of Manchester, 1994.