91Ó°ÊÓ

A differential equation is a mathematical equation that relates a function with its derivatives. In simple terms, it involves an unknown function and its various order derivatives. Differential equations are used to describe a wide range of phenomena such as population growth, motion of waves, heat conduction, and fluid flow.

In the context of this exercise, we are dealing with the equation \( y' = -0.45y \). This is a first-order linear differential equation because it involves the first derivative of \( y \). The equation is of the form \( y' = ky \), where \( k \) is a constant. Such equations are typical when modeling exponential growth or decay processes.

Differential equations can provide valuable insights into how systems evolve over time, and solving them helps us predict the behavior of complex systems.

An initial condition is a value that specifies the state of a system at the beginning of a problem. When solving differential equations, initial conditions are crucial as they allow us to find the specific solution that fits a particular scenario out of the infinite possibilities.

For example, in the exercise, we have the initial condition \( y(0) = -1 \). This tells us that the value of the function \( y \) when \( t = 0 \) is \(-1\). By substituting this initial condition into the general solution of the differential equation, we can determine the specific constant that tailor-makes the solution to our situation.

Understanding and applying initial conditions is essential for accurate predictions in real-world applications, such as predicting the decay of a substance or the cooling of an object.

The exponential growth/decay formula is a mathematical expression used to model processes where quantities increase or decrease at rates proportional to their current value. The general form of the equation is:

• \( y(t) = Ce^{kt} \)

Where:

• \( y(t) \) is the quantity at time \( t \)
• \( C \) is the initial quantity
• \( e \) is the base of the natural logarithm
• \( k \) is the rate constant

If \( k > 0 \), the function represents exponential growth, meaning the quantity increases over time. Conversely, if \( k < 0 \), it depicts exponential decay, indicating the quantity decreases as time progresses.

In this exercise, we identified \( k = -0.45 \), indicating an exponential decay. The initial condition \( y(0) = -1 \) helps us find the specific constant, leading to the solution \( y(t) = -1e^{-0.45t} \). This formula is crucial in fields ranging from biology to economics for modeling systems that grow or shrink exponentially.