Verify that the given function is a solution to the given...

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid. $$y(x)=c_{1} e^{x}+c_{2} e^{-2 x}, \quad y^{\prime \prime}+y^{\prime}-2 y=0$$.

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.
$$y(x)=c_{1} e^{x}+c_{2} e^{-2 x}, \quad y^{\prime \prime}+y^{\prime}-2 y=0$$.

Key Concepts

Maximum Interval of Validity

This concept refers to the largest interval on the independent variable over which the solution is defined and continuous, and within which the differential equation and its solution can be legitimately related. For linear equations with constant coefficients, the interval of validity is typically all real numbers unless initial conditions or singular points impose restrictions.

Verification of Solutions

Verification involves substituting the proposed solution into the differential equation to ensure that it satisfies the equation for all values of the variable within the specified domain. This process confirms the correctness of the solution by demonstrating that it identically satisfies the differential equation.

Exponential Functions in Differential Equation Solutions

Exponential functions often emerge as solutions to linear differential equations with constant coefficients. This happens because when an exponential function is differentiated, the result is a constant multiple of the original function, which simplifies the process of finding a characteristic polynomial.

Linear Homogeneous Differential Equations with Constant Coefficients

These are differential equations in which all terms involve the unknown function or its derivatives, and the coefficients are constants. The structure of such equations allows the use of characteristic equations to find exponential functions as possible solutions, making the analysis and solution process systematic.

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