Problem 27 - Calculus

Question

Find a function \(y(x)\) that solves the differential equation. \(d^{2} y / d x^{2}=1\)

Step-by-Step Solution

Answer

The solution is \( y(x) = \frac{x^2}{2} + C_1x + C_2 \).

Step 1: Understand the equation

The equation given is a second-order differential equation: \( \frac{d^2 y}{dx^2} = 1 \). This means we need to find a function \( y(x) \) such that its second derivative with respect to \( x \) equals \( 1 \).

Step 2: Integrate the equation once

Integrate the equation \( \frac{d^2 y}{dx^2} = 1 \) with respect to \( x \) to find the first derivative \( \frac{dy}{dx} \).\[ \frac{dy}{dx} = \int 1 \, dx = x + C_1 \]where \( C_1 \) is the constant of integration.

Step 3: Integrate the first derivative

Integrate \( \frac{dy}{dx} = x + C_1 \) to find \( y(x) \).\[ y(x) = \int (x + C_1) \, dx = \frac{x^2}{2} + C_1x + C_2 \]where \( C_2 \) is another constant of integration.

Step 4: Write the general solution

The general solution to the differential equation \( \frac{d^2 y}{dx^2} = 1 \) is \( y(x) = \frac{x^2}{2} + C_1x + C_2 \), where \( C_1 \) and \( C_2 \) are constants.

Key Concepts

Second-Order Differential Equation

A second-order differential equation involves the second derivative of a function. This derivative gives us information about how the rate of change of the rate of change occurs—which, in simpler terms, relates to acceleration. In our original exercise, the differential equation is \( \frac{d^2 y}{dx^2} = 1 \). Here, it tells us that the acceleration, or the second derivative of the function \( y(x) \), is constant and equal to 1. Differential equations of this kind are typical in physics, such as in the motion of particles under uniform forces. Understanding the concept of these equations is crucial because it allows us to model real-world phenomena mathematically.

Integration

Integration is the mathematical process used to reverse differentiation. When we are given a derivative and need to find the original function, we use integration. In our exercise, the first integration step transforms the given second derivative \( \frac{d^2 y}{dx^2} = 1 \) into its first derivative \( \frac{dy}{dx} = x + C_1 \), where \( C_1 \) is a constant of integration. This step requires integrating with respect to \( x \), which means finding a function whose derivative gives us conditions included in the problem statement. Here are some key points:- **Indefinite Integration**: The result of integration without limits, including a constant of integration.- **Antiderivative**: Another name for the integral, referring to "reversing" differentiation. The goal of integration here is to progressively find \( y(x) \) by starting with its known acceleration.

Constant of Integration

The constant of integration, represented as \( C_1 \) or \( C_2 \), is a vital aspect of indefinite integration. During the integration process, when a function is differentiated, any constant disappears. Therefore, when integrating, we must include the constant(s) to represent all possible original functions.Upon finding the first derivative, integrating again introduces another constant. This is evident as our original exercise progresses:- After first integration: \( \frac{dy}{dx} = x + C_1 \)- After second integration: \( y(x) = \frac{x^2}{2} + C_1x + C_2 \)Including these constants ensures that we account for all possible original scenarios. These constants are often determined using initial conditions or additional constraints that might be provided in practical problems.

General Solution

The general solution of a differential equation includes terms for unknown constants, which represent a family of all possible solutions that meet the differential condition. In the original exercise, the general solution is given as:\[ y(x) = \frac{x^2}{2} + C_1x + C_2 \]This solution includes two constants, \( C_1 \) and \( C_2 \), because we performed two integrations. Key points for understanding general solutions:- **Complete Set of Solutions**: Includes every possible function that satisfies the differential equation.- **Constants of Integration**: Indicate the solution is general, needing specific conditions to find particular solutions. To find specific solutions, additional conditions, such as initial values or boundaries, can be applied. These would allow determining exact values for \( C_1 \) and \( C_2 \). Hence, the general solution is crucial for representing the entire solution scope unless further constraints are given.