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A separable differential equation is a specific type of differential equation where the two variables can be separated on different sides of the equation. This means you can express the equation in the form \( \frac{dy}{dx} = g(x)h(y) \). By "separable," we mean that you can rearrange it so that one side only involves \( y \) and its derivatives and the other side only involves \( x \). This separation is beneficial for solving the differential equation by integration.

To solve a separable equation like \( \frac{dy}{dx} = \frac{y}{x} \), you would rearrange to get \( \frac{dy}{y} = \frac{dx}{x} \). Then, integrate both sides:

• \( \int \frac{1}{y} \, dy = \int \frac{1}{x} \, dx \)
• The results are \( \ln|y| = \ln|x| + C \),

where \( C \) is the constant of integration. Exponentiating both sides gives \( y = Cx \), aligning with what the equation suggests as the family of solutions (like "(d)": \( y = kx \), where \( k \) can be any constant).

This method is efficient for solving many first-order differential equations where variables can be isolated and integrated separately.

An integrating factor is a technique mainly used to solve linear first-order differential equations that cannot be easily separated. These equations are usually in the form \( \frac{dy}{dx} + P(x)y = Q(x) \). The integrating factor, which is often denoted as \( \mu(x) \), is obtained by computing \( e^{\int P(x) \, dx} \).

This factor simplifies the differential equation, making it solvable by converting it into an exact equation. For instance, take equation (c) from the problem: \( \frac{dy}{dx} = ky + \frac{y}{x} \). Rearranging, it becomes \( \frac{dy}{dx} - ky = \frac{y}{x} \). The coefficient of \( y \) would suggest an integrating factor of \( e^{\int -k \, dx} \), which is \( e^{-kx} \).

Multiplying the whole equation by this integrating factor can transform it into one that can be integrated easily:

• The equation simplifies to a form \( \frac{d}{dx}[\mu(x)y] = \mu(x)Q(x) \).

By integrating both sides, you can solve for \( y \). Thus, with the correct integrating factor, complex differential equations become easier to manage and solve.

First-order linear differential equations form a class of equations that are linear with respect to the variable \( y \) and its first derivative. These equations generally take a standard form:

• \( \frac{dy}{dx} + P(x)y = Q(x) \).

The beauty of first-order linear differential equations lies in their systematic method of solution. This includes the use of integrating factors, as discussed previously, or in some cases, separation of variables if applicable.

For example, consider the differential equation \( \frac{dy}{dx} = ky \), known as the exponential growth equation. It can be solved by easily separating variables and integrating on both sides:

• the integration process leads to \( y = Ce^{kx} \),

reflecting phenomena like population growth or radioactivity decay.

Moreover, first-order differential equations are foundational, as they set the stage for solving more complex higher-order equations and systems used in numerous fields such as physics, engineering, and economic modeling.