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To solve a linear delay differential equation, such as $y'(t) - ay(t-\tau)=0$, use the steps: 1. Solve the corresponding characteristic equation, $\lambda - ae^{-\lambda\tau}=0$. 2. Find the roots of the characteristic equation. 3. Write the general solution using these roots, which might involve exponential functions or oscillatory components depending on the nature of the roots. Functional solution: $y(t) = Ce^{\lambda t}$, where $\lambda$ is a root of the characteristic equation.

Solving a nonlinear delay differential equation, like $y'(t) = f(y(t), y(t-\tau))$, involves: 1. Analyzing the function $f$ for possible simplifications or known forms. 2. Employing numerical methods or finding a fixed point for iterative solutions. 3. Utilizing perturbation techniques or constructing a series solution in cases where an analytical solution is not feasible.

Solving state-dependent delay differential equations where the delay $\tau = g(y(t))$ requires: 1. Understanding of the delay function $g$ and how it changes with the state. 2. Adapting standard methods for constant delays to account for state-dependent changes. 3. Using advanced numerical methods to handle the changing delays as solutions progress. Functional solutions are typically not available in closed form.

For a system of delay differential equations like \begin{align*} y'(t) &= f(y(t), z(t-\tau)), \\ z'(t) &= g(y(t-\tau), z(t)), \end{align*} you should: 1. Investigate each equation separately and the interaction between delays. 2. Typically use numerical solvers designed for systems with delays. 3. Study the stability and bifurcations of the system. Functional solutions for systems are rarely available and heavily dependent on the specific functions and interactions.

To address a neutral delay differential equation such as $y'(t) - ay'(t-\tau)=f(y(t))$, the approach is: 1. Determine the characteristic equation considering the derivative term with delay. 2. Analyze stability and find the neutral stability curves. 3. Depending on the specific equation, either find an analytical solution or resort to numerical approximation. Note: Analytical solutions might be inherently more complex or infeasible.

A distributed delay differential equation involves an integral, such as $y'(t) - a\int_{t-\tau}^t y(s)ds = 0$. To solve it: 1. Analyze the kernel of the integral (e.g., exponential, constant) and simplify the equation. 2. Transform the equation if necessary (e.g., Laplace transform). 3. Seek a functional or numerical solution depending on the complexity. Functional solutions may involve integral transforms or be expressed in terms of special functions.