91Ó°ÊÓ

First-order linear differential equations are a fundamental topic in calculus and differential equations. These equations involve derivatives of a function and can be expressed in the general form: \\[ a(t)\frac{d y}{dt} + b(t)y = c(t) \] \where \( y \) is a function of \( t \), and \( a(t), b(t), \) and \( c(t) \) are functions of \( t \). A characteristic feature of these equations is that they contain only the first derivative of the function, hence "first-order." \
The goal is to find the function \( y(t) \) that satisfies this relationship. First-order linear differential equations are crucial because they model a variety of real-world phenomena, such as population growth, radioactive decay, and cooling processes. The linearity in these equations lies in the fact that both the function and its derivative appear to the first power, making them more straightforward to solve compared to nonlinear differential equations.

The integrating factor is a powerful technique for solving first-order linear differential equations. It transforms the equation into a format that allows straightforward integration. In general, the integrating factor \( \mu(t) \) is calculated using the expression: \\[ \mu(t) = e^{\int \frac{b(t)}{a(t)} \, dt} \] \In the exercise, the integrating factor is specifically calculated as \( (t+1)^2 \). \
Applying the integrating factor involves multiplying every term in the differential equation by \( \mu(t) \). This operation converts the left-hand side into the derivative of the product, \( \frac{d}{dt}(\mu(t)y) \), which can then be easily integrated. \
The integrative power of this technique lies in its ability to rearrange and simplify differential equations, making them solvable through straightforward integration. It’s particularly useful for equations that, at first glance, may not seem easily integrable.

The general solution of a differential equation provides the formula that encompasses all possible specific solutions to the equation. This includes constants of integration introduced during the solution process. For first-order linear differential equations, the solution will typically include terms representing both the particular solution and the homogeneous solution. \
In the given exercise, the general solution is: \\[ s = t+1 + \frac{\ln|t+1|}{(t+1)^2} + \frac{C}{(t+1)^2} \] \where \( C \) is the arbitrary constant which could take any real value, representing an infinite family of solutions. \
This general solution is obtained after simplifying, integrating, and dividing through by the integrating factor. The \( C \) term ensures that any additional conditions or constraints imposed on the original problem, such as initial values, can be accommodated, giving the particular solution for those specific scenarios. Understanding the general solution involves grasping how different terms in the solution portray various aspects influenced by the original differential equation.