Problem 70 - Thomas Calculus in SI Units

Question

Which of the following graphs shows the solution of the initial value problem $$\frac{d y}{d x}=-x, \quad y=1 \text { when } x=-1 ?$$ Give reasons for your answer.

Step-by-Step Solution

Answer

The solution graph is a downward-facing parabola with a vertex at (0, 1.5) matching the point (-1, 1).

Step 1: Identify the Differential Equation

The given initial value problem is \( \frac{dy}{dx} = -x \) with the initial condition \( y = 1 \) when \( x = -1 \). This is a first-order differential equation with a straightforward form, suggesting that integration will be needed.

Step 2: Integrate the Differential Equation

Integrate the differential equation \( \frac{dy}{dx} = -x \) to find \( y \). This yields \( y = \int -x \, dx = -\frac{x^2}{2} + C \), where \( C \) is the integration constant.

Step 3: Apply the Initial Condition

Use the initial condition \( y = 1 \) when \( x = -1 \) to solve for \( C \). Substitute into the equation: \( 1 = -\frac{(-1)^2}{2} + C \). Simplifying gives \( 1 = -\frac{1}{2} + C \), so \( C = 1 + \frac{1}{2} = \frac{3}{2} \).

Step 4: Write the Particular Solution

Incorporate the calculated value of \( C \) into the integrated function. The particular solution to the initial value problem is \( y = -\frac{x^2}{2} + \frac{3}{2} \).

Step 5: Determine Which Graph Corresponds to the Solution

Analyze the shapes or characteristics of the given graphs based on the derived solution. The graph of \( y = -\frac{x^2}{2} + \frac{3}{2} \) is a downward-opening parabola with vertex at \( (0, \frac{3}{2}) \). Check which graph matches these features, especially ensuring the point \((-1, 1)\) is on the graph.

Key Concepts

First-Order Differential Equations

A first-order differential equation is a relation involving an unknown function and its first derivative. It does not involve higher order derivatives. In this exercise, we are dealing with the equation \( \frac{dy}{dx} = -x \).
This type of equation is quite common. It can often be solved by using integration. The term 'first-order' signifies that the highest derivative present is the first derivative, \( \frac{dy}{dx} \).
This means that we are trying to determine how the function changes with respect to \(x\).

• To solve it, we need to integrate the equation.
• The result of integrating gives us a family of functions, or a general solution.
• Using the initial condition, we find a specific solution, also called a particular solution.

The goal is to find the function \(y\) that satisfies both the differential equation and that touches the specified initial condition point (in this exercise, \( y=1 \) when \( x=-1 \)).

Integration Techniques

Integration is a crucial technique for solving differential equations. Here, our task is to integrate the equation \( \frac{dy}{dx} = -x \). The integration gives \( y = -\frac{x^2}{2} + C \), where \( C \) is the integration constant. To understand this better, let us consider the following points:

• Integration can be thought of as the reverse of differentiation.
• The constant \( C \) accounts for all possible vertical shifts of the basic curve.
• The resulting integrated equation represents a family of curves on a graph.

In some cases, like here, an initial condition is given to specifically determine \( C \). With \( y = 1 \) when \( x = -1 \), we place these values into the integrated equation to find \( C \). This ensures that the solution perfectly passes through the initial condition point on the graph.

Graphical Representation of Functions

Graphical representation involves plotting the function on a coordinate plane, showing how the function behaves visually. Let's consider the function \( y = -\frac{x^2}{2} + \frac{3}{2} \), which is derived from solving the differential equation.
Understanding its graph is key to recognizing the right choice among given options. The function is a parabola that opens downwards, since the coefficient of \( x^2 \) is negative. More specific points to remember:

• The vertex of this downward parabola is at \((0, \frac{3}{2})\), which is the highest point.
• The graph is symmetric about the y-axis because it involves \( x^2 \).
• It passes through the point \((-1, 1)\) due to the initial condition.

Visualizing this function helps in identifying which of the graphs provided matches the solution. Analyzing features like the vertex position and the curvature will lead to the correct graph, especially as it should pass through defined points based on the initial value conditions.