Solve each differential equation and graph the function...

Solve each differential equation and graph the function $y=f(x)$ and the polygonal approximation on the same axes. (The approximations were found in Exercises $21-24 . )$.
$$\frac{d y}{d x}=y ; \quad y(0)=1$$

Key Concepts

Ordinary Differential Equation

An ordinary differential equation (ODE) is an equation that involves a function and its derivatives. In the context of this problem, the ODE describes how y changes with respect to x using a first-order derivative, representing a continuous dynamical process.

Initial Value Problem

An initial value problem (IVP) involves solving a differential equation together with a specified value of the function at a given point. The initial condition y(0)=1 singles out a unique solution from the family of all possible solutions to the differential equation.

Analytical Solution Techniques

Analytical solution techniques allow for the derivation of an explicit formula for the unknown function. For first-order separable differential equations, like the one given, methods such as separation of variables lead to closed-form solutions that express y directly in terms of x.

Exponential Growth

Exponential growth refers to processes in which the rate of change of a quantity is proportional to its current value. The equation y' = y is the standard model for exponential growth, and its solution, y = e^x, embodies this idea through the exponential function.

Numerical Approximation

Numerical approximation involves techniques that estimate the solution to a differential equation using discrete steps. Methods such as polygonal approximations or Euler's method construct a piecewise linear graph of the solution, which is useful for visualizing and understanding the behavior of the exact solution.

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