Particular and General solution of Differential equation

Particular solution (The solution for the particular inhomogeneity in question)

In general, an inhomogeneous differential equation is of the form:

$$y'' + p(t)y' + q(t)y = f(t)$$

where $y$ is the unknown function, $p(t)$ and $q(t)$ are known functions, and $f(t)$ is a known function as well. 

The particular solution of an inhomogeneous differential equation is a solution that satisfies the equation for the given function $f(t)$, but with zero initial conditions. In other words, it is a solution that takes into account the effect of the non-homogeneous term $f(t)$, and it is usually denoted by $y_p$.

Digression regarding the zero initial condition criteria

In the context of differential equations, initial conditions refer to the values of the unknown function and its derivatives at some initial time, usually denoted by $t_0$. For example, if we have a second-order differential equation of the form:

$$y'' + p(t)y' + q(t)y = f(t)$$

with initial conditions $y(t_0) = y_0$ and $y'(t_0) = y'_0$, then we are looking for a solution $y(t)$ that satisfies the differential equation and the initial conditions.

However, when we are looking for the particular solution of an inhomogeneous differential equation, we are only concerned with finding a solution that satisfies the equation for the given function $f(t)$. We do not need to satisfy any initial conditions because the general solution of the differential equation already includes the solution to the homogeneous part of the equation, which takes care of the initial conditions.

Therefore, when we find the particular solution, we assume that the function satisfies zero initial conditions, meaning that its value and derivative at some initial time $t_0$ are both zero. This allows us to focus on finding a solution that satisfies only the non-homogeneous part of the equation, without worrying about any initial conditions.

How to find the particular solution

To find the particular solution, one can use a variety of methods depending on the specific form of $f(t)$ and the differential equation. Some common methods include:

- Method of undetermined coefficients: This method involves guessing a particular solution based on the form of $f(t)$, and then solving for the coefficients that make the solution work. For example, if $f(t)$ is a polynomial, one can guess a particular solution of the form $y_p = a_nt^n + a_{n-1}t^{n-1} + \cdots + a_1t + a_0$, and then solve for the coefficients $a_n, a_{n-1}, \ldots, a_0$ by plugging $y_p$ into the differential equation.

- Variation of parameters: This method involves finding a general solution to the homogeneous differential equation (i.e., setting $f(t) = 0$), and then assuming a particular solution of the form $y_p = u(t)y_1(t) + v(t)y_2(t)$, where $y_1(t)$ and $y_2(t)$ are two linearly independent solutions to the homogeneous equation, and $u(t)$ and $v(t)$ are functions to be determined. Plugging this form into the differential equation and solving for $u(t)$ and $v(t)$ will give the particular solution.

- Laplace transform: This method involves taking the Laplace transform of both sides of the differential equation, solving for $Y(s)$ (the Laplace transform of $y(t)$), and then using inverse Laplace transform to find $y(t)$. This method can be particularly useful for linear differential equations with constant coefficients. (Similarly, even fourier transform can be used to find the particular solution).

These are just a few methods that can be used to find the particular solution of an inhomogeneous differential equation. The choice of method depends on the specific equation and the properties of the non-homogeneous term $f(t)$.

General solution ( The solution to the homogenous part)

The general solution of a differential equation is the set of all possible solutions to the equation, including any arbitrary constants that may be present. In other words, the general solution is a family of functions that satisfies the differential equation, but each function in the family may differ from each other by a constant.

For example, consider the first-order linear differential equation:

$$y' + p(t)y = q(t)$$

The general solution to this equation is given by:

$$y(t) = Ce^{-\int p(t) dt} + y_p(t)$$

where $C$ is an arbitrary constant and $y_p(t)$ is the particular solution to the equation (which depends on the function $q(t)$).

The general solution can be obtained by solving the differential equation without specifying any initial conditions. The solution will typically contain one or more arbitrary constants, whose values can be determined by applying initial or boundary conditions.

In some cases, the general solution may involve a family of functions that cannot be expressed in a closed form, but rather as a parametric equation or a series expansion. In any case, the general solution encompasses all possible solutions to the differential equation, and provides a complete description of the behavior of the system governed by the equation.