Question

Solving a Differential Equation In Exercises \(1-10\) , solve the differential equation. $$ \frac{d y}{d x}=x+3 $$

Short answer

The solution of the given differential equation is \( y = \frac{1}{2}x^2 + 3x + c\).

Step-by-step solution

Integrate both sides of the equation

First, the equation \(\frac{d y}{d x} = x + 3 \) needs to be integrated with respect to \(x\) on both sides. This results in \(\int \frac{d y}{d x} dx = \int (x + 3) dx\).

Simplify the integration

\(\int \frac{d y}{d x} dx\) is simply \(y\). And \(\int (x + 3) dx\) becomes \( \frac{1}{2}x^2 + 3x + c\), where \(c\) is the constant of integration. Putting them together we get the integrated equation \( y = \frac{1}{2}x^2 + 3x + c\).

Write the final form of the solution

The final solution of the given differential equation in the general form is \( y = \frac{1}{2}x^2 + 3x + c\).

Key concepts

Integration

Integration is a fundamental concept in calculus, often considered the reverse process of differentiation. When we have a differential equation, like \( \frac{d y}{d x} = x + 3 \), integration helps us to find the original function \( y \). This process involves determining an antiderivative of the function given on the right side of the equation.

To integrate \( \frac{d y}{d x} = x + 3 \), we integrate both sides with respect to \( x \). The left side simplifies to \( y \), because integrating \( \frac{d y}{d x} \) just "undoes" the differentiation. On the right side, we compute \( \int (x + 3) \, dx \).

Why is Integration Important?

• It helps to reverse differentiation, allowing us to find the original function.
• Integration is crucial to solve differential equations.
• Applications span across physics, engineering, economics, and more.

The integral of a simple term \( x \) becomes \( \frac{1}{2}x^2 \), and the constant 3 integrates to \( 3x \). So, the complete integration yields \( \frac{1}{2}x^2 + 3x + c \), where \( c \) is a constant.

Constant of Integration

The constant of integration, often represented by \( c \), plays an important role in calculus. When performing an indefinite integral, there are infinitely many solutions, differing by a constant. This stems from the fact that the derivative of a constant is zero.

In our exercise, when we solved \( \frac{d y}{d x} = x + 3 \), the integration led to \( y = \frac{1}{2}x^2 + 3x + c \). Here, \( c \) represents the constant of integration.

Understanding the Constant of Integration:

• It reflects the infinite set of functions that can share the same derivative.
• The constant ensures every potential solution from the integration process is covered.
• In a geometric sense, it accounts for vertical shifts in the curve of the function.

The exact value of \( c \) can be determined if you have an additional condition or initial value given. Without this extra information, \( c \) remains arbitrary.

General Solution

In the context of differential equations, the general solution is a form that includes all possible solutions. It often incorporates arbitrary constants that can be specified if additional conditions are applied.

When you solve a differential equation like \( \frac{d y}{d x} = x + 3 \), the result, \( y = \frac{1}{2}x^2 + 3x + c \), is a general solution.

Significance of the General Solution:

• The general solution is universal and includes every particular solution possible by adjusting the constant \( c \).
• It provides a broad view, allowing flexibility for particular cases or initial conditions.
• A complete understanding ensures the capability to apply the solution to various problems effectively.

By having a general solution, you open the door to solving the equation for specific scenarios by inserting given values or conditions, and this versatility makes it powerful in mathematical and practical fields.