91Ó°ÊÓ

A linear first-order differential equation is a type of differential equation that can be written in the form: \( y' + P(x)y = Q(x) \). This form reveals two components,\( P(x) \) and \( Q(x) \), which are both functions of \( x \). These equations are called 'first-order' because the highest derivative is the first derivative \( y' \), and 'linear' because \( y \) and its derivatives appear to the power of one.
To solve these equations:

• First, rearrange or transform the equation into the standard linear form, if necessary.
• Identify \( P(x) \), the coefficient of \( y \).
• Identify \( Q(x) \), the standalone term after rearranging.

In our example, by dividing the equation \( xy' + y + x = e^x \) by \( x \), it transforms into \( y' + \frac{1}{x}y = \frac{e^x}{x} - 1 \). Here, \( P(x) = \frac{1}{x} \) and \( Q(x) = \frac{e^x}{x} - 1 \). This is the crucial set-up step for solving the differential equation.

The integrating factor is a powerful tool for solving linear first-order differential equations. It transforms the equation into a form where it can be easily integrated. The formula for the integrating factor \( \mu(x) \) is given by:\[ \mu(x) = e^{\int P(x) \, dx}.\]This factor is designed specifically to make the left side of the equation a derivative of some product, simplifying the process greatly.

In our example:

• We identified \( P(x) = \frac{1}{x} \).
• The integral of \( P(x) \) is \( \int \frac{1}{x} \, dx = \ln|x| \).
• Thus, the integrating factor \( \mu(x) \) becomes \( e^{\ln|x|} = |x| \).

Since we assumed \( x \) is positive, \( |x| \) simplifies to \( x \). Multiplying the entire differential equation by this integrating factor helps transition it into a form that can be simplified using integration.

The product rule is a fundamental concept from calculus used to compute the derivative of the product of two functions. The rule states:\[ \frac{d}{dx}(uv) = u'v + uv',\]where \( u \) and \( v \) are functions of \( x \), and \( u' \) and \( v' \) are their respective derivatives. Understanding this rule is essential for recognizing the structure of certain differential equations that can be written as a derivative of a product.

In the context of our example,

• The left side of the equation becomes \( (xy)' \) after multiplying by the integrating factor \( x \).
• This means that \( xy' + y = \frac{d}{dx}(xy) \), a key realization for integrating both sides directly.

This property of converting into a derivative of a product significantly simplifies the integration process, allowing the whole right-hand side to be integrated straightforwardly, leading to solving the equation.