Verify that the functions y1 and y2 are solutions of the...

verify that the functions $y_{1}$ and $y_{2}$ are solutions of the given differential equation. Do they constitute a fundamental set of solutions?
$$
y^{\prime \prime}-2 y^{\prime}+y=0 ; \quad y_{1}(t)=e^{t}, \quad y_{2}(t)=t e^{t}
$$

Key Concepts

Fundamental Set of Solutions

A fundamental set of solutions for a differential equation is a pair (or a set, for higher order) of solutions that are linearly independent. For second order linear homogeneous differential equations, two linearly independent solutions are needed to construct the general solution through their linear combination. Their independence ensures that they span the entire solution space of the equation.

Second Order Linear Homogeneous Differential Equations

This concept refers to differential equations that have the form y'' + p(t)y' + q(t)y = 0, where the equation involves the function and its derivatives, and there is no non-homogeneous (forcing) term. These equations have the property that any linear combination of solutions is also a solution, which makes finding a complete set of independent solutions particularly useful for constructing the general solution.

Verification by Substitution

To verify that a given function is a solution to a differential equation, the function and its derivatives are substituted into the equation. When the substitution simplifies to an identity (typically 0 = 0 for homogeneous equations), it confirms that the function satisfies the equation. This approach directly checks the validity of the proposed solution.

Wronskian and Linear Independence

The Wronskian is a determinant constructed from solutions and their derivatives, and it is a tool used to test the linear independence of solutions. Specifically, if the Wronskian is non-zero at some point within the interval of interest, the solutions are linearly independent, which is crucial for confirming that a set of solutions forms a fundamental set for the differential equation.

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