Question

Use the method of isoclines to sketch the approximate integral curves of each of the differential equations. \(y^{\prime}=\frac{y}{x^{2}}\)

Short answer

In order to approximate the integral curves of the differential equation \(y'=\frac{y}{x^2}\) using the method of isoclines, we first find isoclines by setting the equation equal to a constant slope, \(c\): \(y'=c \Rightarrow y=cx^2\). Then, we sketch the isoclines for different values of \(c\), which will be either horizontal lines along the x-axis, upward-opening, or downward-opening parabolas. Finally, we draw tangent lines with slope \(c\) on each isocline and approximate the integral curves by connecting tangent lines for all isoclines. The integral curves follow these parabolas, moving from one quadrant to another, giving us an approximate sketch of the solution of the given differential equation.

Step-by-step solution

Identify the differential equation and rearrange if needed

We are given a first-order differential equation: \(y'=\frac{y}{x^2}\). The equation is already in the form \(y'=f(x, y)\), so we won't need to rearrange it.

Solve for different values of the slope

Next, we will find the isoclines by considering constant slope values, say constant \(c\), and solving \(y' = c\). In this case, we will have: \[\frac{y}{x^2} = c\] Now, solve for \(y\): \[y = cx^2\]

Sketch the isoclines

We will now sketch the isoclines for different constant values \(c\). 1. For \(c = 0\), the isocline will be \(y = 0\), which is a horizontal line along the x-axis. 2. For \(c > 0\), the isoclines will be upward-opening parabolas with vertices lying along the x-axis. The greater the value of \(c\), the steeper the parabolas. 3. For \(c < 0\), the isoclines will be downward-opening parabolas with vertices lying along the x-axis. The smaller the value of \(c\), the steeper the parabolas.

Sketch the integral curves on the isoclines

On each isocline, we will draw tangent lines with slope \(c\). Then, we approximate the integral curves by "connecting" tangent lines for all isoclines. 1. For the isocline with \(c = 0\), we draw horizontal tangent lines along the x-axis, which means y is constant and the solutions are of the form \(y = k\) where \(k\) is a constant. 2. For isoclines with \(c > 0\) (upward-opening parabolas), the integral curves will move from the second quadrant toward the first quadrant. The closer the integral curve is to the x-axis, the tighter it follows the parabolas. 3. For isoclines with \(c < 0\) (downward-opening parabolas), the integral curves will move from the third quadrant toward the fourth quadrant. The closer the integral curve is to the x-axis, the tighter it follows the parabolas. Finally, combining all the integral curves will give us an approximate sketch of the solution of the given differential equation \(y'=\frac{y}{x^2}\).

Key concepts

Differential Equations

A differential equation is a mathematical equation that involves an unknown function and its derivatives. These equations are fundamental in describing various phenomena in engineering, physics, biology, and many other fields. They help model how changes in one variable affect another.
In simple terms, it tells you how a function changes at any point. In our exercise, we were given the differential equation \(y' = \frac{y}{x^2}\). Here, \(y'\) is the derivative of \(y\) concerning \(x\), expressing the rate at which \(y\) changes as \(x\) varies.

• First-order Differential Equations: These involve only the first derivative of the function. Our equation \(y' = \frac{y}{x^2}\) is such an example.
• Relevance: They are essential in modeling real-life phenomena like cooling temperature, population growth, or radioactive decay.

The goal in working with these equations is to find a function or a set of functions that satisfy the given equation. The solutions, known as integral curves, provide insight into how the system changes over time.

Integral Curves

Integral curves are the solutions to differential equations. When you solve a differential equation, you are essentially finding these integral curves. They represent the paths traced in a plane by solutions of the differential equation as initial conditions change.
In the context of our exercise involving the differential equation \(y' = \frac{y}{x^2}\), the integral curves are derived from the isoclines we plotted. These curves illustrate the behavior and trend of solutions over the given domain.

• Characteristic Paths: These paths give us a visual representation of how the solution behaves over time.
• Isocline Method: This involves finding lines (isoclines) where the slope \(c\) is constant. From these slopes, tangent lines can be drawn to form the complete solution, which are the integral curves.
• Application: By observing these curves, one can draw conclusions about the system's behavior across different regions of the graph.

Integral curves are essential because they provide valuable information on how systems modeled by differential equations evolve, offering insights into both theoretical and practical situations.

Slope Fields

Slope fields, also known as direction fields, are visual aids to understanding differential equations. They consist of many small line segments or arrows drawn at various points on the plane, each with a slope corresponding to the slope given by the differential equation at that point.
In the exercise, you would use slope fields to visually represent the differential equation \(y' = \frac{y}{x^2}\). By plotting these fields, you get an idea of how integral curves might look even without solving the equation explicitly.

• Construction: For each point \((x, y)\) on the graph, you compute \(y'\) and draw a small line segment with this slope.
• Visualization: Slope fields provide an immediate visual approximation of solutions which makes complex differential equations more intuitive.
• Educational Tool: Particularly useful for students beginning with differential equations, simplifying the understanding of solutions by observation.

However, while slope fields give a good qualitative understanding, they do not provide precise solutions. They act as a guide to how solutions behave across different points on the plane.