Question

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

Short answer

To sketch the approximate integral curves for the differential equation \(y' = xy\), first find the isoclines by solving \(y = \frac{c}{x}\) for different constant slopes \(c\). Then, plot these isoclines on a graph, and sketch the integral curves by drawing tangent lines to the isoclines with slopes equal to the chosen constant \(c\). Analyze the behavior of the integral curves to observe how the solutions to the differential equation change based on the initial conditions.

Step-by-step solution

Identify the isoclines

To find the isoclines, we need to set the given differential equation equal to a constant value. In this case, we have \(y' = xy = c\), where \(c\) is the constant slope. Now solve for the variable \(y\): \[y = \frac{c}{x}\] This equation represents the isoclines of the given differential equation.

Sketch the isoclines

Plot the isoclines for a few different values of the constant slope \(c\). Some commonly used values are \(c = -2, -1, 0, 1, 2\). By varying \(c\), we will get different isoclines; plot these on the same graph. Remember that each isocline represents locations in the plane where the solutions of the differential equation have the same slope.

Sketch the integral curves

Now, using the isoclines as a guide, sketch the approximate integral curves for the given differential equation. To do this, start at a point on an isocline, and draw a curve that is tangent to that isocline, which should have a slope equal to the constant value we chose (i.e., \(c\)). Continue drawing the curve, making sure it remains tangent to the isoclines as you move along. The curves you draw should transition smoothly from one isocline to the next, and they should not intersect each other. Repeat this process for various starting points on different isoclines to get an idea of how the integral curves look.

Analyze the behavior of the integral curves

By looking at your sketch of the integral curves, try to get a general sense of the behavior of the solutions to the given differential equation. Note how the curves approach, intersect, or move away from each other or from the isoclines. In this case, we can observe that as \(x\) increases, the integral curves grow steeply and as \(x\) decreases, the integral curves approach the horizontal axis asymptotically. This gives us an idea of how the solutions to the differential equation behave and change based on the initial conditions.

Key concepts

Integral Curves

Integral curves are solutions to a differential equation displayed visually on a graph. They show how the function behaves over different values of the variables involved.
To draw integral curves, you should:

• Start at a particular initial condition or starting point on the graph.
• Trace along the path suggested by the differential equation, ensuring your path or curve follows the slope at each point.
• Make sure the curve smoothly continues, tangent to the isoclines or slope lines.

These curves give you a visual representation of the behavior of solutions to the differential equation, revealing how the system will evolve over time for different initial conditions. By sketching these curves, you get an insight into how solutions to your differential equation look without having to solve it analytically.

Differential Equations

Differential equations are mathematical equations that involve derivatives, which describe how a function changes. They are used to model real-world phenomena such as motion, growth, decay, and other changes in nature or technology.
In the context of the exercise, we are given a first-order differential equation: \[ y' = xy \]To solve a differential equation, means to find a function or set of functions that satisfy the equation. Differential equations like these tell you the rate of change of a variable, which is why they are fundamental in modeling dynamic systems. By sketching and analyzing these, you can predict the future behavior of the modeled system based on its current state.

Slope Fields

A slope field, or direction field, is a graphical representation used to visualize the solutions of a differential equation. Imagine a grid on a plane where each point has an associated slope, given by the differential equation.
Here's how you use slope fields:

• At each point in the grid, draw a small line or vector with the slope specified by the differential equation.
• Lines should show the direction the solution curves will follow.
• The field acts like a map that guides the drawing of integral curves.

Slope fields are incredibly useful educational tools because they provide a way to approximate the integral curves, helping you get a sense of the equation's behavior without exact calculations. By observing the slope field, you can infer general patterns and dynamics of the equation’s solutions.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) involve functions and their derivatives. They relate the function with its rate of change and are 'ordinary' because they have derivatives with respect to one variable, usually time or space.
ODEs often model systems where change occurs continuously. In the exercise, our task involves an ODE:\[ y' = xy \]ODEs can be of various orders, indicated by the highest derivative, and can be solved using various methods including analytical solutions, numerical approximations, or qualitative analysis like sketching slope fields and integral curves.
Understanding ODEs is key in fields like physics, engineering, and biology because they can model anything from planetary motion to population dynamics, giving insights on how systems respond over time to different initial or boundary conditions.