Question25 Consider the homogeneous linear second-order differential equation with constant coefficients:[math]where [math], [math], and [math] are constants and [math]. Which of the following statements is true about the set of all solutions to this differential equation?Select one alternative: The set of all solutions does not form a vector space because the differential equation involves second derivatives. The set of all solutions does not form a vector space because constant functions are never solutions. The set of all solutions forms a vector space only if [math] and [math]. The set of all solutions forms a vector space because sums and scalar multiples of solutions are also solutions. ResetMaximum marks: 1 Flag question undefined

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BlackTom AI · Accepted Answer

Let's parse the problem carefully and consider what it means for a set of functions to form a vector space in the context of a homogeneous linear differential equation.

Option 1: 'The set of all solutions does not form a vector space because the differential equation involves second derivatives.' This is a misconception. The order of the differential equation does not prevent the solution set from being closed under addition and scalar multiplication. The defining property of a vector space is about combining solutions, not about the order of derivatives in the equation.

Option 2: 'The set of all solutions does not form a vector space because constant functions are never solutions.' This is false in general. Depending on the coefficients, constant functions can be solutions (for example, if the equation is y'' = 0, then constant functions satisfy it). The assertion that constants are never solutions is not guaranteed and is not a valid reason to deny a vector space structure.

Option 3: 'The set of all solutions forms a vector space only if b = 0 and c = 0.' This is overly restrictive. For a homogeneous linear ODE with constant coefficients y'' + b y' + c y = 0, the solution space is still a vector space for any real values of b and c. The dimension of the solution space is determined by the order of the equation (here 2), not by forcing b or c to be zero. Thus this conditional statement is incorrect.

Option 4: 'The set of all solutions forms a vector space because sums and scalar multiples of solutions are also solutions.' This captures the essential linear-algebraic property: if y1 and y2 solve the homogeneous linear ODE, then for any scalars α and β, α y1 + β y2 also solves the ODE due to linearity and homogeneity. The zero function is a solution as well, ensuring the V space axioms hold. Therefore, the solution set is a vector space regardless of the specific values of b and c (as long as the equation remains homogeneous).

In summary, the core idea is that homogenous linear differential equations with constant coefficients form a linear subspace of the function space: closed under addition and scalar multiplication, containing the zero function, and having dimension equal to the order of the equation (here 2). The reasoning above shows why the first three statements are incorrect and why the fourth correctly identifies the vector-space structure of the solution set.

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