Question

Let \(\Phi\) be a solution of $$ Y^{\prime}=A Y+G(t) $$ where \(G\) has a period \(\omega>0\). Prove that the system has a unique solution with period \(\omega\) if and only if \(Y^{\prime}=A Y\) has no solution with period \(\omega\) except \(Y=0\).

Short answer

Question: Prove that a linear first-order differential equation system has a unique solution with period ω if and only if the homogeneous part of the system has no nonzero solution with period ω. Answer: A linear first-order differential equation system has a unique solution with period ω if and only if the homogeneous part of the system has no solution with period ω except Y=0. This is proven by considering two cases: (1) if the homogeneous part has no solution with period ω except Y=0, the system has a unique solution with period ω, and (2) if the system has a unique solution with period ω, then the homogeneous part has no solution with period ω except Y=0.

Step-by-step solution

Proving that if the homogeneous part has no solution with period ω except Y=0, the system has a unique solution with period ω.

Assume that the homogeneous part Y'=AY has no solution with period ω except Y=0. Let's now examine the inhomogeneous part Y'=AY+G(t) where G(t) has period ω. Let Y₁(t) and Y₂(t) be two solutions of Y'=AY+G(t) with period ω. We have: $$Y₁^{\prime}(t) = AY₁(t)+G(t) \quad (1)$$ $$Y₂^{\prime}(t) = AY₂(t)+G(t) \quad (2)$$ Subtract equation (1) from equation (2): $$Y₁^{\prime}(t) - Y₂^{\prime}(t) = A(Y₁(t) - Y₂(t))$$ Let Y(t) = Y₁(t) - Y₂(t), we have: $$Y^{\prime}(t) = AY(t)$$ Y(t) has a period ω, but since we assumed that the homogeneous part has no solution with period ω except Y=0, we must have Y(t)=0. This implies that Y₁(t) = Y₂(t), meaning there is a unique solution with period ω for the inhomogeneous part.

Proving the converse: if the system has a unique solution with period ω, then the homogeneous part has no solution with period ω except Y=0.

We now assume that the inhomogeneous part Y'=AY+G(t) has a unique solution with period ω. Suppose that we have a nontrivial solution of the homogeneous part Y'=AY with period ω, denoted by Y₃(t). Let Y₄(t) be the unique solution of the inhomogeneous equation. Then, we can observe that: $$Y₅(t) = Y₄(t) + Y₃(t)$$ is also a solution of the inhomogeneous equation with period ω: $$Y₅^{\prime}(t) = Y₄^{\prime}(t) + Y₃^{\prime}(t) = AY₄(t) + AG(t) + AY₃(t) = AY₅(t) + G(t)$$ But, we assumed that the system has a unique solution with period ω, which means Y₅(t) = Y₄(t). This implies Y₃(t) = 0, showing that the homogeneous part has no solution with period ω except Y=0. Therefore, the system has a unique solution with period ω if and only if the homogeneous part Y'=AY has no solution with period ω except Y=0.

Key concepts

Ordinary Differential Equations

An ordinary differential equation (ODE) is an equation involving a function and its derivatives. The function usually represents a physical quantity while the derivatives as rates of change. When an ODE is expressed as \( Y' = f(t, Y) \), it is depicting how the rate of change of \(Y\) is dependent on t (time) and \(Y\) itself.

ODEs can be classified based on their linearity into homogeneous and inhomogeneous types. A homogeneous ODE's right-hand side consists solely of terms that depend on the function Y and its derivatives, while an inhomogeneous ODE includes additional terms, like \( G(t) \), which are functions of t only. The problem in the exercise involves finding periodic solutions to an inhomogeneous ODE where \( G(t) \) has a specific period \( \omega \).

Inhomogeneous Differential Equations

An inhomogeneous differential equation includes terms that do not depend on the unknown function \( Y \) or its derivatives. These are given functions of the independent variable, t, like \( G(t) \) in our problem. For the equation \( Y' = AY + G(t) \), \( G(t) \) represents the inhomogeneous component.

Periodic Inhomogeneity

When \( G(t) \) is periodic with period \( \omega \) such that \( G(t + \omega) = G(t) \), it imparts a periodic nature to the differential equation. The solution to such an equation can attain a form that repeats itself after every \( \omega \), becoming a periodic solution, under certain conditions as exemplified in the homework problem.

Uniqueness Theorem

The uniqueness theorem for ODEs, often referred to as the Existence and Uniqueness Theorem, states that under certain conditions, an initial value problem has a unique solution. This theorem applies to both linear and nonlinear ODEs.

The uniqueness is crucial for our exercise since it asserts that, for the given differential equation with periodic inhomogeneity, if the homogeneous counterpart has no non-zero periodic solutions, then the inhomogeneous equation will have a unique periodic solution.

Homogeneous Differential Equations

Contrasting with inhomogeneous, a homogeneous differential equation only contains terms with the function \( Y \) or its derivatives. The standard form is \( Y' = AY \), without any additional functions of t.

In our problem, the existence of a zero-only periodic solution in the homogeneous equation \( Y'=AY \) guarantees the uniqueness of the periodic solution for the inhomogeneous equation. This is because the presence of any non-zero periodic solution in the homogeneous equation would indicate the possibility of similarly periodic solutions in the inhomogeneous equation, thereby contradicting uniqueness.