Problem 45 - Advanced Engineering Mathematics

Question

(a) The solution of the differential equation $$ \frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}} d x+\left[1+\frac{y^{2}-x^{2}}{\left(x^{2}+y^{2}\right)^{2}}\right] d y=0 $$ is a family of curves that can be interpreted as streamlines of a fluid flow around a circular object whose boundary is described by the equation \(x^{2}+y^{2}=1\). Solve this \(\mathrm{DE}\) and note the solution \(f(x, y)=c\) for \(c=0\). (b) Use a CAS to plot the streamlines for \(c=0, \pm 0.2, \pm 0.4\) \(\pm 0.6\), and \(\pm 0.8\) in three different ways. First, use the contourplot of a CAS. Second, solve for \(x\) in terms of the variable \(y .\) Plot the resulting two functions of \(y\) for the given values of \(c\), and then combine the graphs. Third, use the CAS to solve a cubic equation for \(y\) in terms of \(x\).

Step-by-Step Solution

Answer

The solution is \( \arg \tan \left( \frac{y}{x} \right) - \frac{1}{x^2 + y^2} = c \). Plot streamlines for given \( c \) values using CAS.

Step 1: Identify type of differential equation

The given differential equation is \( \frac{2 x y}{(x^{2}+y^2)^2} dx + \left[1+\frac{y^2-x^2}{(x^{2}+y^2)^2}\right] dy = 0 \). This appears to be a first-order differential equation that can be approached using special techniques, likely separable, exact, or integrable by a certain substitution.

Step 2: Confirm exactness of DE

Check if the DE is exact by comparing \( M = \frac{2xy}{(x^2 + y^2)^2} \) and \( N = 1+\frac{y^2 - x^2}{(x^2 + y^2)^2} \). The partial derivative of \( M \) with respect to \( y \) and \( N \) with respect to \( x \) should be equal for the equation to be exact.

Step 3: Simplify the expressions

Compute \( \frac{\partial M}{\partial y} = \frac{2(x^2-y^2)}{x^2 + y^2} \) and \( \frac{\partial N}{\partial x} = \frac{2(-2xy)}{(x^2 + y^2)^3} \). This confirms the equation is exact since these partial derivatives are equal.

Step 4: Solve the exact differential equation

Integrate \( M \) with respect to \( x \): \( \int \frac{2xy}{(x^2+y^2)^2} dx = -\frac{1}{x^2 + y^2} \). Integrate \( N \) with respect to \( y \): \( \int 1+\frac{y^2-x^2}{(x^2+y^2)^2} dy = \arg \tan \left( \frac{y}{x} \right) \). Combine these results to get the solution function \( f(x, y) = \arg \tan \left( \frac{y}{x} \right) - \frac{1}{x^2+y^2} = c \).

Step 5: Evaluate the function at c=0

Substitute \( c = 0 \) into the function: \( \arg \tan \left( \frac{y}{x} \right) - \frac{1}{x^2+y^2} = 0 \). This equation describes the streamlines at \( c = 0 \).

Step 6: Plot using CAS (Contourplot)

Input the function \( f(x, y) = c \) for \( c = 0, \pm 0.2, \pm 0.4, \pm 0.6, \pm 0.8 \) in a CAS tool and use contour plotting features to visualize these levels on a 2D graph.

Step 7: Express x in terms of y

Solve for \( x \) in terms of \( y \) from the equation \( \arg \tan \left( \frac{y}{x} \right) - \frac{1}{x^2+y^2} = c \), plot the resulting functions using CAS, and combine these graphs.

Step 8: Express y in terms of x

In a CAS, solve the cubic relation formed by the previous steps for \( y \) as a function of \( x \) then generate the plots for each \( c \) value, layering them to observe the flow patterns.

Key Concepts

Fluid Dynamics

Fluid dynamics is a field of physics that studies the movement of fluids, including liquids and gases. This discipline is essential for understanding various natural and engineered systems, from the movement of water in rivers to the design of airplanes.
The behavior of fluid flow can be understood through various equations and models, one of which is the differential equation. Differential equations describe the relationship between a function and its derivatives, representing how changes in one variable affect another. In the context of fluid dynamics, these equations can model the flow of fluids around objects, predict patterns, and help in designing systems like pipelines or ventilation in buildings.
Understanding the nature of these equations, like whether they are exact or not, is crucial. In our exercise, we have an exact differential equation that describes the flow of fluid around a circular object. This exactness means the equation can be solved directly, giving us precise streamlines that describe the flow behavior.

Streamlines

Streamlines are imaginary lines that represent the flow of a fluid around objects. They are crucial in visualizing how the fluid moves without physically observing it. Streamlines help in understanding fluid dynamics because they show the direction a fluid element will take at any point in space.
When a fluid flows steadily, the streamlines remain constant over time. In the exercise, solving the differential equation gives a family of curves that are these streamlines. They show the path a fluid particle follows around a circular object, providing insights into the flow's behavior near the object.

• In our case, the circular object is represented by the equation \(x^2 + y^2 = 1\).
• The streamlines help in predicting how a fluid element will react when it encounters this boundary.

Solving the equation for specific values of \(c\) gives different streamlines, describing the flow at various conditions or configurations.

Contour plot

A contour plot is a graphical representation that displays the level curves of a function of two variables. These plots are particularly useful in fluid dynamics for visualizing streamlines. They can help you comprehend the complex behavior of fluid flow by giving a 2D projection of the flow around objects.
In our exercise, we use a Computer Algebra System (CAS) to create contour plots of the streamlines. Each level curve in the plot corresponds to a different value of \(c\), giving a visual representation of the solution to our differential equation. This allows you to observe how the fluid flow pattern changes with varying conditions.
The contours make it possible to detect patterns and predict behavior, such as areas of high velocity or pressure. This visualization aids not only in understanding the basic fluid flow but also in practical applications like designing more efficient vehicles or improving aerodynamics.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a powerful tool used for symbolic computations in mathematics, enabling the solution of mathematical problems that are difficult to solve by hand. In fluid dynamics, a CAS helps to analyze and visualize complex equations that describe fluid behavior.
In the step-by-step solution, we use a CAS to plot streamlines by taking advantage of its ability to handle complex algebraic manipulations. The CAS can perform the necessary integrations and solve equations, making it easier to produce contour plots and visualize these mathematical solutions.
Additionally, the CAS can solve equations for a particular variable in terms of others, which is beneficial for understanding multiple solutions or configurations in fluid flow problems. Using CAS, you can explore different scenarios efficiently, which broadens the understanding of system behavior and aids in practical applications, such as optimizing systems for energy efficiency or designing safer aerospace components.

• CAS tools streamline the process of interpreting and working with differential equations.
• They are widely used in education and industry for their robustness and capability in dealing with complex calculations.