integraalfaktori

An integraalfaktori is a mathematical tool used to solve first-order linear ordinary differential equations. These equations are generally of the form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x. The goal of using an integrating factor is to transform this equation into a form where both sides can be directly integrated with respect to x.

The integrating factor, often denoted by $\mu(x)$, is calculated as $e^{\int P(x) dx}$. This factor is then multiplied by every term in the differential equation. The key property of the integrating factor is that when multiplied by the left side of the equation, dy/dx + P(x)y, it creates the derivative of a product. Specifically, $\mu(x)(dy/dx + P(x)y)$ becomes equal to $d/dx(\mu(x)y)$.

Once the equation is multiplied by the integrating factor, it takes the form $d/dx(\mu(x)y) = \mu(x)Q(x)$. Since the left side is now a perfect derivative, it can be integrated directly. Integrating both sides with respect to x yields $\mu(x)y = \int \mu(x)Q(x) dx + C$, where C is the constant of integration. Finally, to solve for y, the equation is divided by the integrating factor $\mu(x)$, giving the general solution $y(x) = (1/\mu(x))(\int \mu(x)Q(x) dx + C)$. This method provides a systematic way to find the solution to a specific class of differential equations.