Final Answer:
The explicit general solution to the given differential equation, y, is:
y = Ce^{x}
Explanation:
In this differential equation, (y), represents the dependent variable, and (x) is the independent variable. The solution (Ce^{x}) is an exponential function, where (C) is an arbitrary constant. To derive this solution, let's analyze the given differential equation:
y' = {dy}/{dx}
In our case, (y) itself is equal to (y'), meaning the derivative of (y) with respect to (x) is simply (y). So, the differential equation becomes:
y = Ce^{x}
To verify this solution, we can take the derivative of \(y\) with respect to \(x\):
{dy}/{dx} = Ce^{x}
This indeed equals (y), confirming that (y = Ce^{x}) is a solution to the differential equation. The constant (C) allows for a family of solutions, encompassing various initial conditions or scenarios.
In conclusion, the explicit general solution to the given differential equation is an exponential function (Ce^{x}), where (C) is an arbitrary constant. This solution satisfies the differential equation, as evidenced by the equality of (y) and its derivative (y'). The inclusion of the constant (C) enables the solution to accommodate different starting conditions or specific scenarios within the broader family of solutions.