91Ó°ÊÓ

An integrating factor is a function that helps us solve first-order linear differential equations. To understand it, consider the general form of the equation:
\[ y' + P(t)y = Q(t), \] our goal is to find a function, called the integrating factor, that simplifies this equation.
The formula to find the integrating factor, \( \mu(t) \), is: \[ \mu(t) = e^{\int P(t) dt}. \]
Let's break that down:

• First, identify \(P(t)\) from the standard form of the differential equation.
• Second, compute the integral of \(P(t)\).
• Finally, raise e to the power of the result from the integral.

In the exercise, we identified \(P(t) = \sqrt{t}\).
Taking the integral of \(\sqrt{t}\) gives us \( \frac{2}{3} t^{3/2}.\)
So, the integrating factor is: \[ \mu(t) = e^{\frac{2}{3} t^{3/2}}.\] The role of \(\mu(t)\) is to convert the differential equation into a form where it can be easily integrated.

Differential equations involve derivatives, which represent rates of change. A first-order differential equation, like the one in the exercise, includes the first derivative of the function.
The general form of a first-order linear differential equation is
\[ y' + P(t)y = Q(t), \]
where \( y' \) represents the derivative of \( y \) with respect to \( t \), \( P(t) \) is a given function of \( t \), and \( Q(t) \) is another function of \( t \).

• These equations can model various phenomena such as population growth, radioactive decay, and cooling of objects.
• First-order indicates that only the first derivative (\( y' \)) appears in the equation.

Understanding how to identify and manipulate the components of these equations is crucial for solving them.
In the exercise, we transformed the given equation into its standard form. By identifying \(P(t)\) as \( ∑\sqrt{t} \) and \(Q(t)\) as \(2(t+1)\), we used these components to find the integrating factor.

Integration is the process of finding the antiderivative of a function. It is a fundamental concept in calculus used to solve differential equations.
The power rule for integration states that for any function \(t^n\):

• \[ \int t^n dt = \frac{t^{n+1}}{n+1} + C,\]

where \(n e -1\) and \(C\) is the constant of integration.
In the given exercise, we computed the integral of \( ∑\sqrt{t} \), which is \( t^{1/2} \). Using the power rule, we find: \[ \int t^{1/2} dt = \frac{2}{3} t^{3/2}. \]This integral is then used to find the integrating factor: \[ \mu(t) = e^{\frac{2}{3} t^{3/2}}. \]Integration enables us to accumulate values and find the reverse process of differentiation. By integrating the function \( \sqrt{t}\), we ensure that our equation can be manipulated into a more manageable form.