Use the concept that y=k on (-∞, ∞) is a constant function if and only if d y / d x=0 to determi

step 1 In this question, we're going to use the concept that if y is equal to k on minus infinity to in

Use the concept that y=k on (-∞, ∞) is a constant function...

Use the concept that $y=k$ on $(-\infty, \infty)$ is a constant function if and only if $d y / d x=0$ to determine whether the given differential equation possesses constant solutions. Solve the given differential equation. Assume that $k$ is a real number. $x \frac{d y}{d x}+6 y=18$

Key Concepts

Constant Function

A constant function, defined as y = k for all x in its domain, is characterized by a derivative equal to zero. This property allows us to test differential equations for constant solutions by setting the derivative term to zero and observing if a corresponding constant value satisfies the equation.

Constant Solution Test

When testing an equation for constant solutions, one substitutes y = k and dy/dx = 0 into the differential equation. The resulting algebraic equation must hold for all x in the domain, thereby indicating the existence of a constant solution. This basic method is useful for quickly identifying trivial or equilibrium solutions in differential equations.

First Order Linear Differential Equation

A first order linear differential equation is an equation of the form dy/dx + P(x)y = Q(x). Such equations can often be solved using standard techniques, and they typically have both homogeneous and particular solutions. Identifying the structure of these equations is crucial to applying the appropriate solution methods.

Integrating Factor Method

The integrating factor method is a systematic technique for solving first order linear differential equations. It involves multiplying the entire equation by an integrating factor, usually calculated as exp(?P(x) dx), which transforms the left-hand side into the derivative of a product. Once this transformation is done, the equation can be integrated directly, leading to a general solution that includes an arbitrary constant, thereby accounting for initial or boundary conditions.

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