Determine whether the functions satisfy the deferential equation. y^''-4 y=0 ; y1(x)=e^2 x,

step 1 In this question, we are given the differential equation. Y prime prime minus 4Y is equal to 0.

Determine whether the functions satisfy the deferential...

Key Concepts

Linear Homogeneous Differential Equations

A linear homogeneous differential equation is an equation where the dependent variable and its derivatives appear linearly and the equation is set equal to zero. In these types of equations, the principle of superposition applies, meaning that any linear combination of solutions is also a solution. This allows finding a general solution composed of independent solutions, often through methods like the characteristic equation.

Characteristic Equation

For constant coefficient linear differential equations, one can often assume solutions of the form e^(rx) which leads to a polynomial equation in r known as the characteristic equation. The roots of this polynomial determine the form of the general solution. In the context of the equation y'' - 4y = 0, the characteristic equation is r^2 - 4 = 0, which yields roots that inform the structure of the solution.

Verification of Candidate Solutions

Verifying that a function is a solution to a differential equation involves computing its derivatives and substituting them into the equation to check if the equation is satisfied (i.e., it reduces to an identity such as 0=0). This method is especially straightforward for linear homogeneous equations with constant coefficients, where the structure of the functions, like exponentials, often makes the verification process direct.

Hyperbolic Functions and Their Relation to Exponentials

Hyperbolic functions, such as sinh and cosh, are defined in terms of exponential functions. For example, sinh(x) is defined as (e^x - e^(-x))/2. This relationship allows hyperbolic functions to also serve as solutions to differential equations with constant coefficients, because their underlying exponential form implies they satisfy similar derivative relationships as exponentials.

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