Problem 2 - Advanced Engineering Mathematics

Question

From \(d y=(x+1)^{2} d x\) we obtain \(y=\frac{1}{3}(x+1)^{3}+c\).

Step-by-Step Solution

Answer

Integrating \( dy = (x+1)^2 dx \) gives \( y = \frac{1}{3}(x+1)^3 + c \), verifying the solution.

Step 1: Understand the Differential Equation

The given equation is a differential equation of the form \( dy = (x+1)^2 dx \). Our task is to find the function \( y \) by integrating the right side with respect to \( x \).

Step 2: Set Up the Integral

To solve for \( y \), we integrate both sides of the equation with respect to \( x \). This gives us \( y = \int (x+1)^2 \, dx \).

Step 3: Perform the Integration

To perform the integration \( \int (x+1)^2 dx \), we can expand the integrand as \( (x+1)^2 = x^2 + 2x + 1 \). So, the integral becomes \( \int (x^2 + 2x + 1) \, dx \).

Step 4: Integrate Each Term Separately

Integrate each term of the polynomial separately:- The integral of \( x^2 \) is \( \frac{1}{3}x^3 \).- The integral of \( 2x \) is \( x^2 \).- The integral of \( 1 \) is \( x \).Thus, \( \int (x^2 + 2x + 1) \, dx = \frac{1}{3}x^3 + x^2 + x + C \).

Step 5: Finish the Solution

Now substitute back the expression we expanded for \( (x+1)^2 \) and compare it to the original form \( \int (x+1)^2 \, dx \). In our context, this returns to the format \( y = \frac{1}{3}(x+1)^3 + c \), where \( C \) is an arbitrary constant.

Key Concepts

Differential Equations

A differential equation is an equation that relates a function with its derivatives. In essence, it describes the rate of change and how different factors affect this change. In everyday life, you can think of it as a recipe that needs to be followed to determine how a particular quantity evolves over time or other dimensions.
For example, in the problem we started with, the differential equation is given by:

• \( dy = (x+1)^2 \, dx \)

This equation states that the infinitesimal change in \( y \) (denoted by \( dy \)) is equal to \((x+1)^2\) times the infinitesimal change in \( x \) (denoted by \( dx \)). To solve for \( y \), we need to integrate the equation, which involves reversing the differentiation process.Differential equations are powerful tools used in a variety of disciplines such as physics, engineering, and economics, as they serve to model complex systems and predict their behaviors.

Polynomial Integration

Polynomial integration is a technique used to find the antiderivative or integral of polynomial expressions. It essentially asks, "What function, when differentiated, will give us this polynomial back?"
In our problem, when solving:

• \( \int (x+1)^2 \, dx \)

we first expand the expression \((x+1)^2\) into a polynomial:

• \( x^2 + 2x + 1 \)

The next step is to integrate each term separately. For each term in the polynomial, use the power rule for integration:

• For \( x^n \), it becomes \( \int x^n \, dx = \frac{1}{n+1} x^{n+1} \)

Applying this rule:

• The integral of \( x^2 \) results in \( \frac{1}{3}x^3 \)
• The integral of \( 2x \) results in \( x^2 \)
• The integral of \( 1 \) results in \( x \)

Combining all these integrations gives:

• \( \int (x^2 + 2x + 1) \, dx = \frac{1}{3}x^3 + x^2 + x + C \)

Arbitrary Constants

After integrating, you will notice the appearance of a constant denoted by \( C \), often referred to as the constant of integration. This arbitrary constant is crucial to the solution of an indefinite integral.
When we differentiate functions, any constant ends up disappearing: for example, \( \frac{d}{dx}(x + 5) = 1 \) and \( \frac{d}{dx}(x + 3) = 1 \). Thus, without specific boundary conditions or initial values, we cannot know what the original constant might have been; hence, it remains arbitrary. In our step-by-step solution,

• \( y = \frac{1}{3}x^3 + x^2 + x + C \)

This arbitrary constant \( C \) essentially accounts for all these unknown constants from the multiple routes that differentiation could have taken. In practical applications, if more information is provided (like an initial condition), you can solve for \( C \) to get a specific value.