91Ó°ÊÓ

In solving first-order linear differential equations, the integrating factor method becomes an essential tool. Here, the purpose of the integrating factor is to transform a differential equation into an exact equation. This means we convert the original equation into one where the left-hand side can be expressed as the derivative of a product.
The integrating factor \( \mu(x) \) for a differential equation of the form \( y' + P(x)y = Q(x) \) is calculated using the formula \( \mu(x) = e^{\int P(x) \, dx} \). This factor simplifies the process of finding the solution.

• To calculate the integrating factor, you first identify \( P(x) \).
• Next, compute the integral of \( P(x) \) to find the exponent for the integrating factor.
• In our example, \( P(x) = \cos(x) \), so the integrating factor is \( e^{\sin(x)} \).

This systematic approach allows us to proceed with transforming and subsequently integrating the original differential equation.

Finding the general solution involves transforming the differential equation to help isolate \( y \). Once an equation is multiplied by the integrating factor, it can often be rewritten as a simple derivative of a product.
This step is crucial because it allows us to integrate directly.

• For the given problem, after multiplying by the integrating factor, the equation becomes \((y \cdot e^{\sin(x)})' = \cos(x) \cdot e^{\sin(x)}\).
• The left side represents the derivative of \( y \cdot e^{\sin(x)} \), allowing for simple integration.
• When integrated over \( x \), both sides yield functions plus an arbitrary constant \( C \).

The general solution describes all possible solutions of the differential equation. In our example: \( y(x) = 1 + Ce^{-\sin(x)} \). This encompasses every curve that satisfies the differential equation given any initial conditions.

The substitution method in solving differential equations is a supportive strategy that is especially useful in handling integrals complicating our work. In our context, substitution eases the integration of the right-hand side of the simplified equation.
The key steps involve:

• Selecting a substitution that simplifies the integral. Here, \( u = \sin(x) \).
• Transforming the differential \( dx \) into \( du \) using the derivative \( du = \cos(x) \, dx \).
• The troublesome integral \( \int \cos(x) \cdot e^{\sin(x)} \, dx \) becomes \( \int e^{u} \, du \), which simplifies significantly.

Finally, upon integrating and substituting back, we solve for \( y \) to reveal the general solution in terms of the original variable \( x \). This substitution streamlines calculations, preventing potential errors and offering a clear path to the solution.