Let α, β∈ℝ. Suppose f, g: ℝ →ℝ are differentiable functions such that f^'=αf+βg, g^'=αg-βf,

step 1 For this problem, we are asked to evaluate f prime of 0 if f of x equals x to the power of alpha

Let α, β∈ℝ. Suppose f, g: ℝ →ℝ are differentiable functions...

Let $\alpha, \beta \in \mathbb{R}$. Suppose $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are differentiable functions such that $$ f^{\prime}=\alpha f+\beta g, \quad g^{\prime}=\alpha g-\beta f, \quad f(0)=0, \quad \text { and } \quad g(0)=1 $$ Show that $f(x)=e^{\alpha x} \sin \beta x$ and $g(x)=e^{\alpha x} \cos \beta x$ for all $x \in \mathbb{R}$. (Hint: Consider $h: \mathbb{R} \rightarrow \mathbb{R}$ given by $h(x):=\left(f(x)-e^{\alpha x} \sin \beta x\right)^{2}+(g(x)-$ $\left.e^{\alpha x} \cos \beta x\right)^{2}$. Find $h^{\prime}$.) (Compare Exercise 6 of Chapter 4.)

Let $\alpha, \beta \in \mathbb{R}$. Suppose $f, g: \mathbb{R} \rightarrow \mathbb{R}$ are differentiable functions such that
$$
f^{\prime}=\alpha f+\beta g, \quad g^{\prime}=\alpha g-\beta f, \quad f(0)=0, \quad \text { and } \quad g(0)=1
$$
Show that $f(x)=e^{\alpha x} \sin \beta x$ and $g(x)=e^{\alpha x} \cos \beta x$ for all $x \in \mathbb{R}$. (Hint:
Consider $h: \mathbb{R} \rightarrow \mathbb{R}$ given by $h(x):=\left(f(x)-e^{\alpha x} \sin \beta x\right)^{2}+(g(x)-$
$\left.e^{\alpha x} \cos \beta x\right)^{2}$. Find $h^{\prime}$.) (Compare Exercise 6 of Chapter 4.)

Key Concepts

Candidate Solution Verification

This technique entails proposing a potential solution for a differential equation or system and then substituting it back into the original equations to verify its validity. In the given problem, one proposes f(x) = e^(?x) sin(?x) and g(x) = e^(?x) cos(?x) and checks that these functions satisfy both the system of differential equations and the initial conditions.

Method of Invariants (Constant of Motion)

The method of invariants involves identifying a function that remains constant along the solutions of the differential equation. In this case, by defining a function h(x) that measures the squared difference between the proposed solution and the actual functions, and showing that its derivative is zero, one concludes that h(x) is constant. Combined with appropriate initial conditions, this results in h(x) being zero for all x, thereby confirming the candidate solution.

System of Linear Differential Equations

This concept involves sets of differential equations in which the derivatives of the unknown functions are expressed as linear combinations of the functions themselves. In the context of the problem, the equations f' = ?f + ?g and g' = ?g - ?f form a linear system whose behavior can often be analyzed using techniques such as matrix exponentiation, eigenvalues, and eigenvectors to understand its dynamics.

Uniqueness Theorem for Initial Value Problems

This fundamental theorem in differential equations asserts that if a solution to an initial value problem exists under appropriate conditions, it is unique. Once the candidate solution is verified to satisfy both the differential equations and the initial conditions, the uniqueness theorem guarantees that it is the only solution, thereby completing the verification process.

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