What is the solution [math: y=f(x)] that all second order homogeneous linear differential equations have in common. Just enter [math: f(x)].

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BlackTom AI · Accepted Answer

To approach this question, start by recognizing what a second-order homogeneous linear differential equation looks like: a y'' + b y' + c y = 0, where a, b, c are constants (or functions of x in a more general form). The term “homogeneous” means the right-hand side is zero, so the zero function is always a candidate solution.

First, consider the zero function y(x) = 0. If you plug it into the equation, its derivatives y' = 0 and y'' = 0, so the left-hand side becomes a(0) + b(0) + c(0) = 0, which satisfies the equation for any choice of a, b, c. Hence y = 0 is a valid solution for every such differential equation.

Next, why isn’t there a nonzero function that is common to all such equations? Because the general solutions to homogeneous second-order equations depend on the specific coefficients a, b, c (or functions a(x), b(x), c(x)). Different equations have different characteristic roots, leading to different solution forms (linear combinations of exponentials, sines/cosines, or polynomials times exponentials). Therefore, only the trivial solution y = 0 is guaranteed to satisfy every possible homogeneous second-order linear differential equation, regardless of its specifics.

In this context, the requested f(x) to enter is the zero function. Since the prompt asks for the solution all such equations share, f(x) = 0, i.e., the zero function, is the universal solution. The statement that every second-order homogeneous linear ODE has only the zero function in common would be too strong; rather, the universal common solution across all such equations is the zero function itself, with nontrivial solutions depending on the particular equation.

What is the solution [math: y=f(x)] that all second order homogeneous linear differential equations have in common. Just enter [math: f(x)].

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