Question

The solution of \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)=0\) represents: (A) a point (B) a straight line (C) a parabola (D) a circle

Short answer

The solution of the given equation \(\frac{d^2y}{dx^2} = 0\) represents a straight line, as integrating twice gives the equation \(y = C_1x + C_2\), which is the equation for a straight line. So, the correct answer is (B) a straight line.

Step-by-step solution

Integrate the given equation once

Given the second derivative of y with respect to x is 0: \(\frac{d^2y}{dx^2} = 0\) Integrate with respect to x: \(\frac{dy}{dx} = \int 0 \, dx = C_1\) Where \(C_1\) is the constant of integration.

Integrate the equation again

Now integrate the equation \(\frac{dy}{dx} = C_1\) with respect to x: \(y = \int C_1 \, dx = C_1x + C_2\) Where \(C_1\) and \(C_2\) are constants of integration.

Identify the shape of the solution

The equation we have obtained is in the form: \(y = C_1x + C_2\) This equation represents a straight line, hence the solution represents a straight line. So, the correct answer is (B) a straight line.

Key concepts

Second Derivative

The second derivative, denoted as \(\frac{d^2y}{dx^2}\), is a crucial concept in calculus that tells us about the concavity and the rate of change of the slope of a function. It's essentially the derivative of the first derivative \(\frac{dy}{dx}\). If \(\frac{d^2y}{dx^2} = 0\), this means that the slope of the function is constant, which indicates a linear relationship. In simple terms, the function has no curvature; it's a straight line. This is precisely why the equation \(\frac{d^2y}{dx^2} = 0\) results in a straight line equation. Constant slope means no acceleration in the curve, aligning with the characteristics of a linear graph.

Integration

Integration is the process of finding the original function from its derivative. When we start with the second derivative equation, like \(\frac{d^2y}{dx^2} = 0\), we integrate it to find back the first derivative. When integrated for the first time, it yields \(\frac{dy}{dx} = C_1\), which basically tells us the slope of the function. Here, \(C_1\) is the integration constant.

After this, we integrate again to retrieve the original function: \(y = C_1x + C_2\). This second integration results in a linear function reflecting a straight line. Each integration step decreases the order of the derivative, bringing us closer to the original function. Choosing the correct constants from geometry or boundary conditions can specify which straight line or shape we're analyzing.

Linear Equations

A linear equation is any equation that can be expressed in the form \(y = mx + b\), where \(m\) and \(b\) are constants. This equation represents a straight line in a two-dimensional space, with \(m\) describing the slope and \(b\) being the y-intercept. From the solution: \(y = C_1x + C_2\), we clearly see a linear equation setup.

This relates to the integration step where \(C_1\) and \(C_2\) are constants. Linear equations play a pivotal role in various applications, including physics and economics, where they help express relationships between quantities. They simplify complex systems into easily interpretable insights, often foundational for higher-level mathematical studies and problem-solving techniques.

Geometric Interpretation

The geometric interpretation of the solution to differential equations helps understand what the equation looks like graphically. For \(\frac{d^2y}{dx^2} = 0\), after solving and simplifying, we get the line equation \(y = C_1x + C_2\). This represents a straight line in geometry. The constants \(C_1\) and \(C_2\) determine the slope and position of the line, respectively.

• When \(C_1 > 0\), the line has a positive slope and rises.
• If \(C_1 < 0\), it declines.
• \(C_2\) shifts the line up or down without affecting the slope.

Understanding that a differential equation solution leads to a particular shape in geometry aids in visualizing mathematical concepts, embedding a tangible component to abstract formulas. This visualization is indispensable in fields like engineering, physics, and environmental science.