Question

Slope Field In Exercises 31 and \(32,\) sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. To print an enlarged copy of the graph, go to MathGraphs.com. $$ \frac{d y}{d x}=-\frac{x}{y} $$

Short answer

The general solution of the differential equation \(\frac{d y}{d x}=-\frac{x}{y}\) is \(y = \pm \sqrt{x^2 + 2C}\). Different specific solutions correspond to different values of the constant C, which could be visualized from slope field.

Step-by-step solution

Separate the Variables

The given differential equation is first to be rewritten in a form that separates the variables x and y. We do this by multiplying both sides by dx, and then cross multiplying : \(-x dx = y dy\).

Integrate Both Sides

Both sides of the equation are integral expressions. Integrate both sides to get rid of the differentials : \(\int -x dx = \int y dy\). This simplifies to \(-0.5x^2 = 0.5y^2 + C\)

Solve for y

Multiply both sides by -2 to isolate y and simplify the general solution. This gives : \(x^2 = -y^2 + 2C\). So, \(y^2 = x^2 + 2C\). The final general solution will be \(y = \pm \sqrt{x^2 + 2C}\)

Key concepts

Slope Field

Imagine you are hiking and have a map that shows how steep the ground is at any given point. A slope field is similar, but for equations instead of hills. It's a graph that shows the slope of the solution to a differential equation at various points in the plane. For the given differential equation \( \frac{d y}{d x}=-\frac{x}{y} \), a slope field would consist of little line segments at several points representing the slope at those points, which is negative \(x/y\). This visual representation is a powerful tool for understanding the behavior of differential equations before solving them analytically.
As you plot these slopes and begin to imagine curves that fit through them, you are seeing possible solutions to the differential equation. Even without the actual solution, the slope field allows you to predict how the function behaves, which is particularly useful for complex equations where finding an exact solution is challenging.

Separation of Variables

The technique known as separation of variables applies to differential equations that can be rearranged to isolate all terms involving one variable (in this case, \(y\)) on one side of the equation and those involving another variable (in this case, \(x\)) on the other side. This is much like sorting the fruits from the vegetables in your grocery bag - each goes to its own place.
In our exercise, we start with \( \frac{d y}{d x}=-\frac{x}{y} \), and through some algebraic manipulation—multiplying by \(dx\) and rearranging—we get \( -x dx = y dy \). Now that the variables are separated with \(x\)'s on one side and \(y\)'s on the other, each side can be integrated separately. This essential technique turns a complex differential equation into a more simple equation we can solve.

Integration

At the heart of calculus is integration, the process of finding the whole from the sum of its parts. It is the reverse of differentiation: if differentiation gives you the slope at any given point (the part), integration gives you the overall shape of the graph (the whole).
For the separated equation \( -x dx = y dy \), we integrate each side to find the function \(y\) that describes our graph. After integration, we obtain \( -\frac{1}{2}x^2 = \frac{1}{2}y^2 + C \), where \(C\) is the constant of integration. This magic step takes us from a world of rates of change, represented by derivatives, into a realm of quantities, represented by the functions themselves. The integration stitches together an infinite number of infinitesimal pieces (the derivatives) to reveal the shape of our function.

General Solution

The general solution of a differential equation is not just one function but a whole family of possible functions that satisfy the equation. It's akin to a family photo, where every member shares common features but each is unique in their own way.
In this case, after separating variables and integrating, we arrive at \( x^2 = -y^2 + 2C \). By solving for \( y \), we get the general solution \( y = \pm \sqrt{x^2 + 2C} \), where \(C\) can be any real number. Each different value of \(C\) gives us a different curve, and all these curves make up the family of solutions that fit our original differential equation. The general solution encompasses all specific solutions and gives us a comprehensive understanding of the possible behaviors of the function described by the differential equation.