91Ó°ÊÓ

A first-order linear ordinary differential equation (ODE) has the general form \( y' + p(x)y = q(x) \) where \( y \) is the unknown function of \( x \) that we’re trying to solve for, \( y' \) is the derivative of \( y \) with respect to \( x \) also known as the rate of change of \( y \), \( p(x) \) is a known function, and \( q(x) \) is another known function.

The solution procedure typically involves finding an integrating factor, which is a function used to multiply both sides of the equation to aid in solving the ODE. The integrating factor generally transforms the left side of the equation into the derivative of a product. This simplifies the equation, allowing us to integrate and find \( y(x) \) explicitly. In the exercise provided, we are dealing with a first-order linear ODE, where \( p(t) = 1/2 \) and \( q(t) = 6 \).

The integrating factor is a powerful method employed to solve first-order linear ODEs. It is denoted as \( IF \) and is calculated using the coefficient of \( y \) from the ODE. The formula for the integrating factor is \( IF = e^{\int p(x)dx} \).

For our exercise, the integrating factor is determined from the coefficient \( \frac{1}{2} \) as \( IF = e^{\int \frac{1}{2} dt} = e^{\frac{t}{2}} \). When we multiply the entire ODE by this \( IF \) we make the left side of the differential equation a perfect derivative of the product of \( IF \) and the unknown function \( z(t) \), which tremendously simplifies the process of solving the equation.

Using the integrating factor transforms the problem into one that can be solved by direct integration, highlighting its usefulness in finding solutions to ODEs.

Separation of variables is yet another technique to solve differential equations, particularly useful when dealing with equations that can be factored into functions of \( y \) and functions of \( x \) independently. The idea is to rearrange the equation so that each variable and its derivatives are on opposite sides of the equation, and then integrate both sides.

This method, however, is only applicable when the differential equation can be manipulated into a separable form. While our given exercise does not specifically require this method, understanding separation of variables is crucial, as it is one of the simplest methods to solve ODEs when possible. In scenarios where equations are not readily separable or are non-linear, integration factors or other methods would need to be employed instead.