Question

The differential equation of family of curves \(\mathrm{y}=\mathrm{Ax}+(\mathrm{B} / \mathrm{x})\) is: (A) \(\left.\mathrm{y}\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)+\mathrm{x}^{2} \mathrm{dy} / \mathrm{dx}\right)-\mathrm{y}=0\) (B) \(y\left(d^{2} y / d x^{2}\right)+x^{2}(d y / d x)+y=0\) (C) \(x^{2}\left(d^{2} y / d x^{2}\right)+x(d y / d x)-y=0\) (D) \(x^{2}\left(d^{2} y / d x^{2}\right)+x(d y / d x)+y=0\)

Short answer

Thus, the correct differential equation of the family of curves \(y = Ax + (B/x)\) is: \(x^2\left(\frac{d^2y}{dx^2}\right) + x\left(\frac{dy}{dx}\right) - y = 0\).

Step-by-step solution

Find the first derivative

First, we'll find the first derivative of \(y\) with respect to \(x\). Given: \[ y = Ax + \frac{B}{x} \] Taking the first derivative with respect to \(x\), we get: \[ \frac{dy}{dx} = A - \frac{B}{x^2} \]

Find the second derivative

Now, let's find the second derivative of \(y\) with respect to \(x\), using the first derivative we found in Step 1: \[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left(A - \frac{B}{x^2}\right) \] Taking the derivative, we get: \[ \frac{d^2y}{dx^2} = \frac{2B}{x^3} \]

Eliminate the arbitrary constants

Let's express \(A\) and \(B\) in terms of other known variables. We can rewrite the equation of the first derivative to express \(A\) as: \[ A = \frac{dy}{dx} + \frac{B}{x^2} \] Then, substituting this expression of \(A\) into the original family of curves equation, we get: \[ y = x\left(\frac{dy}{dx} + \frac{B}{x^2}\right) + \frac{B}{x} \] Now, let's eliminate \(B\). Multiply both sides by \(x^2\), we get: \[ x^2y = x^3\frac{dy}{dx} + Bx + Bx^2 \] Rearranging to get \(B\) in terms of other known variables, we have: \[ B = \frac{x^2y - x^3\frac{dy}{dx}}{x + x^2} \]

Substitute the second derivative and eliminate A and B from the equation

Now, we'll substitute the value of \(B\) we found in Step 3 into the expression for the second derivative: \[ \frac{d^2y}{dx^2} = \frac{2(x^2y - x^3\frac{dy}{dx})}{x^3(x + x^2)} \] Simplify the equation, we have: \[ (x^3 + x^4)\frac{d^2y}{dx^2} = 2(x^2y - x^3\frac{dy}{dx}) \]

Convert to the differential equation

Finally, let's rearrange the equation we derived in Step 4 to get the differential equation form: \[ x^4\frac{d^2y}{dx^2} + x^3\frac{d^2y}{dx^2} - 2x^2y + 2x^3\frac{dy}{dx} = 0 \] Comparing this equation with the given options, we find that it matches option \(C\): \[ x^2\left(\frac{d^2y}{dx^2}\right) + x\left(\frac{dy}{dx}\right) - y = 0 \]

Key concepts

Family of Curves

Understanding the concept of a family of curves is vital in differential equations, as they represent a group of curves that are described by a single general equation with one or more arbitrary constants. In the context of differential equations, identifying a family of curves helps us understand how different solutions of the same differential equation are related to each other.

For instance, the given family in our problem, represented by the equation
\[y = Ax + \frac{B}{x}\]
involves two arbitrary constants, A and B. The solutions to the corresponding differential equation will embody a multitude of curves, each for specific values of A and B. These curves share a common differential property that can be discovered by eliminating the arbitrary constants A and B through differentiation, which is exactly what the solution steps aim to accomplish.

First Derivative

The first derivative of a function provides essential insights into the behavior of the function's graph, such as the slope of the tangent line at any point on the curve, and the increasing or decreasing nature of the function values. In the given problem, the first derivative is calculated as:
\[\frac{dy}{dx} = A - \frac{B}{x^2}\]
This expression reveals how the slope of our family of curves changes with respect to x. It plays a key role in the exploration of the family's properties and the formulation of a differential equation that is free from the constants A and B.

Second Derivative

The second derivative, symbolized as
\[\frac{d^2y}{dx^2}\],
indicates the rate of change of the slope of the curve or, in other words, the concavity of a function’s graph. The second derivative helps to reveal points of inflection, where the concavity changes, and assists in determining the overall shape of the graph. For the given differential equation, the second derivative is:
\[\frac{d^2y}{dx^2} = \frac{2B}{x^3}\]
This result, when used in conjunction with the original family of curves and the first derivative, allows for the elimination of the constants A and B to produce a differential equation that applies to the entire family. This second derivative is a crucial step in characterizing the curvature of the family of curves, leading us closer to the differential equation that governs them.