91Ó°ÊÓ

The integrating factor is a crucial tool when solving linear first-order differential equations. It allows us to transform a non-exact equation into one that is easily integrable. In our problem, the differential equation is

• \( (1+x^2) \frac{\mathrm{d} y}{\mathrm{d} x} + 3xy = 5x \).

Here, we were looking for a way to multiply the entire equation by a function, known as the integrating factor, to simplify the problem. In the general form, this function is found as

• \( \mu(x) = e^{\int \frac{Q(x)}{P(x)} \mathrm{d}x} \).

For our specific problem, with

• \( Q(x) = 3x \)
• \( P(x) = 1+x^2 \),

we calculate the integrating factor to be

• \( (1+x^2)^{3/2} \).

By multiplying the differential equation by this factor, the left side becomes the derivative of a product, making the problem much simpler to integrate.

Initial conditions are necessary when solving differential equations, as they allow us to determine any arbitrary constants that appear during integration. In the context of first-order linear differential equations, like the one in our example, initial conditions ensure we find the unique solution applicable to a particular scenario. For this exercise, the initial condition provided is

• \( y = 2 \) when \( x = 1 \).

This condition helped us solve for the constant \( C \) after integrating the differential equation. When we applied this, it meant substituting \( x = 1 \) and \( y = 2 \) into our integrated equation. By doing so, we calculated

• \( C = -\sqrt{2} \).

This simple condition altered what could have been multiple solutions down to one exact path.

The substitution method is a powerful technique in calculus for dealing with integrals that are difficult to solve directly. In solving for the integrating factor of our given differential equation, substitution played a key role. We encountered the integral

• \( \int \frac{3x}{1+x^2} \mathrm{d}x \),

and by setting a substitution of

• \( u = 1 + x^2 \),

we were able to transform this integral into a simpler form. Through substitution and differentiation

• \( \frac{\mathrm{d}u}{\mathrm{d}x} = 2x \),

we could rewrite the integral, successfully evaluate it, and thus find our integrating factor

• \((1+x^2)^{3/2} \).

This technique often simplifies complex problems, leading to more straightforward solutions.