Problem 49
Question
(a) Sketch the slope field for \(y^{\prime}=x / y\) (b) Sketch several solution curves. (c) Solve the differential equation analytically.
Step-by-Step Solution
Verified Answer
The solution curves follow the equation \( y^2 = x^2 + C \).
1Step 1: Understand the Differential Equation
The equation given is a first-order differential equation: \( y' = \frac{x}{y} \). This means the slope at any point \((x, y)\) is determined by the ratio \(\frac{x}{y}\).
2Step 2: Sketch the Slope Field
To sketch the slope field, compute the slope \(\frac{x}{y}\) at several points \((x, y)\). For instance, at \((1,1)\), the slope is \(\frac{1}{1} = 1\), while at \((0,2)\), the slope is \(\frac{0}{2} = 0\). Continue this for a grid of points to visualize the slope field.
3Step 3: Draw Solution Curves
Using the slope field, draw potential solution curves. These curves are tangent to the slopes in the field and demonstrate the behavior of solutions originating from different points.
4Step 4: Solve the Differential Equation Analytically
Separate the variables to solve the equation: \( y \frac{dy}{dx} = x \). Integrating both sides, \( \int y \, dy = \int x \, dx \), gives \( \frac{y^2}{2} = \frac{x^2}{2} + C \). Simplifying, we get \( y^2 = x^2 + C' \), where \( C' = 2C \).
5Step 5: Express General Solution
The general solution of the equation is \( y^2 = x^2 + C \). This is an implicit solution representing a family of curves.
Key Concepts
Slope FieldSolution CurvesVariable SeparationIntegrationImplicit Solution
Slope Field
A slope field, also known as a direction field, is a visual representation used to depict the solutions of a differential equation graphically. It consists of small line segments or arrows drawn at various points on the coordinate plane. Each segment represents the slope of the solution curve at that particular point.
To construct a slope field for the differential equation \( y' = \frac{x}{y} \), we calculate the slope at multiple chosen points, such as (1,1) and (0,2). At (1,1), the slope would be 1, showing a 45-degree angle; at (0,2), the slope would be 0, indicating a horizontal line. As you plot these lines across a grid, a pattern emerges, illustrating how the slope varies over the plane.
This slope field assists in understanding how solutions to the differential equation might behave without solving it analytically.
To construct a slope field for the differential equation \( y' = \frac{x}{y} \), we calculate the slope at multiple chosen points, such as (1,1) and (0,2). At (1,1), the slope would be 1, showing a 45-degree angle; at (0,2), the slope would be 0, indicating a horizontal line. As you plot these lines across a grid, a pattern emerges, illustrating how the slope varies over the plane.
This slope field assists in understanding how solutions to the differential equation might behave without solving it analytically.
Solution Curves
Solution curves are graphical representations that incorporate the slope field to provide insights into the behavior of solutions. These curves are drawn such that they are tangent to the slope markings at every point they pass through.
Once the slope field is sketched for the equation \( y' = \frac{x}{y} \), you can draw these solution curves through specific initial points. For example, a curve passing through (1,1) will have a slope of 1 at that point. This curve can be extended, maintaining tangency with all subsequent slope indications encountered.
These curves often illustrate how solutions deviate depending on their initial conditions, showcasing the diverse behaviors solutions can exhibit while adhering to the differential equation.
Once the slope field is sketched for the equation \( y' = \frac{x}{y} \), you can draw these solution curves through specific initial points. For example, a curve passing through (1,1) will have a slope of 1 at that point. This curve can be extended, maintaining tangency with all subsequent slope indications encountered.
These curves often illustrate how solutions deviate depending on their initial conditions, showcasing the diverse behaviors solutions can exhibit while adhering to the differential equation.
Variable Separation
Variable separation is a method used to solve differential equations by isolating the terms involving each variable on opposite sides of the equation. This technique is applicable when you can express the equation in the form of \( f(y) \frac{dy}{dx} = g(x) \), which allows both sides to be integrated separately.
For our equation, \( y' = \frac{x}{y} \), we can rewrite it as \( y \frac{dy}{dx} = x \). This allows us to separate the variables, leading to \( y \, dy = x \, dx \).
For our equation, \( y' = \frac{x}{y} \), we can rewrite it as \( y \frac{dy}{dx} = x \). This allows us to separate the variables, leading to \( y \, dy = x \, dx \).
- Multiply both sides by dx to get them in terms of y and x.
- Integrate both sides individually, yielding expressions only in terms of y and x, respectively.
Integration
Integration is a fundamental calculus operation used to find the antiderivative of a function. In the context of differential equations, integration is applied to both sides of the separated equation obtained through variable separation.
With our separated equation \( y \, dy = x \, dx \), we integrate the left side with respect to y and the right side with respect to x:
With our separated equation \( y \, dy = x \, dx \), we integrate the left side with respect to y and the right side with respect to x:
- \( \int y \, dy = \frac{y^2}{2} + C \)
- \( \int x \, dx = \frac{x^2}{2} + C \)
Implicit Solution
An implicit solution is a solution of a differential equation that expresses the relationship between the dependent and independent variables without solving explicitly for one of the variables. It often takes the form of an equation involving both variables.
In our scenario, after integrating, we arrived at the implicit form \( \frac{y^2}{2} = \frac{x^2}{2} + C \). By multiplying through by 2, this simplifies to \( y^2 = x^2 + C \), representing a family of curves.
These curves are the solution paths described earlier as solution curves. The implicit solution is powerful because it encompasses multiple specific solutions depending on the initial conditions, each corresponding to a different value of C.
In our scenario, after integrating, we arrived at the implicit form \( \frac{y^2}{2} = \frac{x^2}{2} + C \). By multiplying through by 2, this simplifies to \( y^2 = x^2 + C \), representing a family of curves.
These curves are the solution paths described earlier as solution curves. The implicit solution is powerful because it encompasses multiple specific solutions depending on the initial conditions, each corresponding to a different value of C.
Other exercises in this chapter
Problem 47
Solve the differential equations in Problems \(44-47 .\) Assume \(x, y, t > 0\). $$\frac{d y}{d t}=-y \ln \left(\frac{y}{2}\right), \quad y(0)=1$$
View solution Problem 49
Give an explanation for your answer. There is a solution curve for the logistic differential equation \(d P / d t=P(2-P)\) that goes through the points (0,1) an
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Give an explanation for your answer. For any positive values of the constant \(k\) and any positive values of the initial value \(P(0),\) the solution to the di
View solution Problem 50
(a) Sketch the slope field for \(y^{\prime}=-y / x\) (b) Sketch several solution curves. (c) Solve the differential equation analytically.
View solution