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A first-order linear differential equation is an equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \). The order refers to the highest derivative present in the equation, which is one in this case. Linear implies that both the function \( y \) and its derivative \( \frac{dy}{dx} \) are linear, or linear-like, meaning they appear to the first power and are not multiplied by each other.

In the problem at hand, \( \frac{dM}{dt} = 0.05M \) is a classic example of such an equation, showcasing the simplicity and elegance of first-order problems. Here, \( P(x) = -0.05 \) and \( Q(x) = 0 \). Solutions typically involve finding an integrating factor, but when they are as straightforward as this one, separating variables and integrating often does the trick too. Thus, first-order linear differential equations can often be tackled using various strategies based on their form.

Separable differential equations are a specific type of differential equation where variables can be separated on different sides of the equation. This trick allows each variable to be integrated separately, simplifying the process of solving the equation.

In the equation \( \frac{dM}{dt} = 0.05M \), it can be reorganized to \( \frac{1}{M} \, dM = 0.05 \, dt \). What makes this separable is the ability to isolate differentials—\( dM \) with \( M \) on one side, and \( dt \) with its constant on the other.

• First, separate the variables: This gives you \( \frac{1}{M} \, dM = 0.05 \, dt \).
• Next, integrate both sides: This can be handled separately for each variable. \( \int \frac{1}{M} \, dM = \ln |M| \), while \( \int 0.05 \, dt = 0.05t \).

Separable equations are popular for their simplicity and clear path to integration, often providing a straightforward route to a solution, as shown in this textbook solution.

The general solution represents a family of solutions to a differential equation that includes arbitrary constants. For anyone studying differential equations, understanding these general solutions is crucial.

In our example, we began with \( \ln |M| = 0.05t + C \), derived from the integration process of separating variables.

• This translates to \( |M| = e^{0.05t + C} \) after exponentiation.
• By setting \( A = e^C \), we simplify to \( M = Ae^{0.05t} \), where \( A \) encompasses the arbitrary constant, integral to the general solution.

The general solution \( M(t) = Ae^{0.05t} \) encapsulates all potential specific solutions by varying \( A \). It recognizes the inherent complexities of initial conditions or boundaries, creating pathways to precise solutions with more contextual information, such as initial values.