Let α∈ℝ and f: ℝ →ℝ be a differentiable function such that f^'=αf and f(0)=1. Show that f(x)=e^α

step 1 For this problem, we have, we want to show that f of x equals alpha times the natural log of x,

Let α∈ℝ and f: ℝ →ℝ be a differentiable function such that...

Let $\alpha \in \mathbb{R}$ and $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}=\alpha f$ and $f(0)=1$. Show that $f(x)=e^{\alpha x}$ for all $x \in \mathbb{R}$. (Compare Exercise 5 of Chapter $4 .$ )

Key Concepts

Ordinary Differential Equations

Differential equations involve relationships between a function and its derivatives. In particular, first-order ordinary differential equations describe the rate of change of a function with respect to one independent variable and often arise in modeling exponential growth or decay processes.

Initial Value Problem

An initial value problem involves finding a solution to a differential equation that satisfies a specified condition at a given point, for example, f(0)=1. This extra piece of information is essential to determine a unique solution among the family of possible solutions of the differential equation.

Uniqueness Theorem for ODEs

The uniqueness theorem guarantees that under appropriate conditions, such as the function being differentiable, a differential equation with a given initial condition has exactly one solution. This theorem is crucial for asserting that the solution f(x) = e^(?x) is the only function that satisfies the differential equation f' = ?f and the initial condition f(0)=1.

Exponential Functions

Exponential functions are fundamental solutions to differential equations of the form f' = ?f. They are characterized by the property that their rate of change at any point is proportional to their current value, making them the natural solution to such problems. In this context, the specific exponential function e^(?x) naturally arises as the unique solution.

Transcript

Please give Ace some feedback