Question

Family \(\mathrm{y}=\mathrm{Ax}+\mathrm{A}^{3}\) of curves is represented by the differential equation of degree: (A) 1 (B) 2 (C) 3 (D) 4

Short answer

The family of curves is given as \(y = Ax + A^3\). We differentiate it with respect to \(x\) to obtain \( \frac{dy}{dx} = A \). Substituting the value of \(A\) back in the original equation, we get \(y = (\frac{dy}{dx}) * x + (\frac{dy}{dx})^3\). The highest power of the derivative present in the equation is 1, so the order of the differential equation is 1. The answer is (A) 1.

Step-by-step solution

Identify the given family of curves

The given family of curves is y = Ax + A^3.

Differentiate the expression with respect to x

Differentiate the given equation concerning x to eliminate the parameter A as follows: Differentiate y = Ax + A^3 concerning x: \( \frac{dy}{dx} = A \)

Solve for the parameter A

Now, we have the value of A in terms of its derivative, which is A = \( \frac{dy}{dx} \).

Substitute the value of A back in the original equation

Substitute the value of A back in the original equation for the given family of curves and rewrite it: y = (dy/dx) * x + (\( \frac{dy}{dx} \))^3

Identify the order of the differential equation

The obtained equation is y = (dy/dx) * x + (\( \frac{dy}{dx} \))^3. Since the highest power of the derivative present in the equation is 1, the order of the differential equation is 1. Therefore, the correct option is: (A) 1

Key concepts

Family of Curves

Understanding the concept of the 'family of curves' is crucial when dealing with differential equations in mathematics. A family of curves refers to a set of curves that are described by a single equation with one or more parameters. In the given exercise,
the family of curves is represented by the equation \( y = Ax + A^3 \), where \(A\) is the parameter. Each value of \(A\) provides a different member of this family, representing various curves on the coordinate plane.

Importance of Parameters

In a family, the parameter(s) accommodate the differences between individual curves. These parameters account for shifts, stretches, or rotations that alter the shape or position of the basic curve.

Relevance to Differential Equations

In the context of differential equations, finding a relation that encompasses the entire family of curves implies eliminating the parameters to obtain a single differential equation. This approach shows how a diverse set of graphs can be unified under an overarching mathematical structure, demonstrating the power of differentiation in understanding complex sets of solutions.

Differentiation

Differentiation is a fundamental operation in calculus that measures how a function changes as its input changes. It's the process of finding the derivative, which is essentially the rate of change or slope of a curve at a particular point.

Application in the Exercise

In our exercise, differentiation is used to remove the parameter \(A\) from the given family of curves. By differentiating \( y = Ax + A^3 \) with respect to \(x\), we obtain \(\frac{dy}{dx} = A \). This is a crucial step since it transforms an implicit family of curves into an explicit differential equation.

Understanding Derivatives

A derivative represents a concept that may seem abstract at first, but it's akin to finding the instantaneous velocity of an object in motion if we think of the graph as a path over time. Differentiation allows students to tackle problems that involve rates of changes, such as growth rates, velocity, and acceleration, broadening their ability to model and solve real-world scenarios.

Order of Differential Equation

The concept of the 'order of a differential equation' relates to the highest derivative taken of a function or the highest order of differential used in the equation. In simpler terms, it describes the number of times you have differentiated a function to get to the particular form of the equation.

Identification in Solutions

To determine the order of a differential equation, we look for the highest derivative that appears in the equation. For instance, if the highest derivative is the second derivative, the order is 2. In the exercise's solution, the highest derivative present is \(\frac{dy}{dx} = A\), which is a first-order derivative. Therefore, the order of the differential equation derived from the given family of curves is 1.

Significance of Order

The order of a differential equation is crucial as it often provides insights into the potential complexity and the methods needed to solve the equation. It directly influences the number of initial conditions required to define a unique solution. Higher-order differential equations can describe more intricate systems, which is why understanding this concept is vital for students delving into advanced mathematics or practical applications like physics, engineering, and economics.