Question

Solve the initial value problem. (1−7)−=0,(4)=−5

Answer 1

In the given initial value problem, the solution is
`\(y(x) = Ce^x - 5\)`, where (C) is the constant determined by the initial condition. Plugging in (x = 4) and (y(4) = -5), we find (C = -7). Therefore, the final solution is
`\(y(x) = -7e^x - 5\)`. This exponential decay model represents the behavior of the system described by the given differential equation.

Explanation:

In solving the provided initial value problem (y'(x) - 7 = 0), with the initial condition (y(4) = -5), the first step involves integrating the differential equation. The general solution to the differential equation is
`\(y(x) = Ce^x\)`, where (C) is a constant. However, to determine the specific solution, we apply the initial condition by substituting (x = 4) and (y(4) = -5) into the general solution. This yields the equation
`\(-5 = Ce^4\)`, and solving for (C) gives (C = -7). Thus, the final answer is
`\(y(x) = -7e^x - 5\)`.

This exponential decay model implies that the rate of change of (y) with respect to (x) is directly proportional to the current value of (y), and the negative sign indicates a decreasing behavior over time. Exponential decay models find application in diverse fields, from describing population decline in biology to analyzing the depreciation of assets in finance.

In this context, the solution provides a mathematical representation of a system undergoing exponential decay, with the constant (C) determined by the initial condition, ensuring the accuracy of the model.