Problem 17 - Thomas Calculus

Question

In Exercises \(13-18\) , find the orthogonal trajectories of the family of curves. Sketch several members of each family. $$ y=c e^{-x} $$

Step-by-Step Solution

Answer

The orthogonal trajectories are parabolas given by \( y^2 = 2x + C \).

Step 1: Identify the differential equation of the given family

We are given the family of curves \( y = ce^{-x} \). To find the differential equation, differentiate both sides with respect to \( x \). The derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = -ce^{-x} \), which simplifies to \( \frac{dy}{dx} = -y \).

Step 2: Substitute and rearrange to find orthogonal trajectories

For orthogonal trajectories, the slope \( \frac{dy}{dx} \) should be the negative reciprocal of the given family. Therefore, set \( \frac{dy}{dx} = \frac{1}{y} \). This is the differential equation whose solution will give the orthogonal trajectories.

Step 3: Solve the differential equation for orthogonal trajectories

We now solve the differential equation \( \frac{dy}{dx} = \frac{1}{y} \). This can be rearranged as \( ydy = dx \). Integrate both sides: Integrating the left side, \( \int y \, dy = \frac{y^2}{2} \), and the right side, \( \int dx = x + C \). Therefore, \( \frac{y^2}{2} = x + C \). Multiply through by 2 to get \( y^2 = 2x + C \).

Step 4: Interpret the solution

The solution \( y^2 = 2x + C \) represents the orthogonal trajectories to the given family \( y = ce^{-x} \). This family of curves is a parabola that opens along the x-axis.

Step 5: Sketch the curves

Sketch the given family \( y = ce^{-x} \) by plotting several curves such as \( y = e^{-x}, 2e^{-x}, 0.5e^{-x} \) which show exponential decay. Then sketch the orthogonal trajectories \( y^2 = 2x + C \) for several \( C \), representing parabolas that intersect the exponential curves at right angles.

Key Concepts

Differential Equations

In mathematics, differential equations are equations that involve unknown functions and their derivatives. They are a fundamental tool used to describe the behavior of complex systems. For instance, they can depict how quantities change over time or space.
Finding orthogonal trajectories can often involve differential equations. The orthogonal trajectories cross the initial family of curves at right angles. To obtain them, we typically use a differential equation and modify it to suit the orthogonal condition.

• Step 1: Differentiate the original equation to find the slope of the curves.
• Step 2: Change the slope by taking its negative reciprocal. This ensures curves are orthogonal.
• Step 3: Solve the new differential equation to find the orthogonal family.

This process was used in the exercise where the exponential function's slope was inverted to yield the orthogonal trajectory's slope. Understanding differential equations is crucial in gaining deeper insights into how these functions interact.

Exponential Functions

Exponential functions demonstrate a consistent growth or decay rate and are prevalent in natural phenomena. They follow the form: \[ y = ce^{kx} \]where \( c \) and \( k \) are constants affecting the growth rate and direction.
In the exercise, you explored the family of curves \( y = ce^{-x} \). Here, the negative exponent \(-x\) leads to exponential decay, meaning the function decreases as \( x \) increases.

• \( c \) controls the initial value at \( x=0 \).
• The decay rate is constant due to the negative exponent.

Exponential functions are critical in various areas, like modeling population growth or radioactive decay. Sketching these curves helps visualize the continuous decrease over increasing \( x \), providing visual insight into their behavior.

Parabolas

Parabolas are classic, symmetrical curves defined by quadratic functions. The general form is:\[ y^2 = 4ax \]where \( a \) determines the parabola's width and direction.
In the exercise, the orthogonal trajectories \( y^2 = 2x + C \) form parabolas that are symmetrical around the x-axis. These parabolas can open to the right or left, depending on the equation.

• The constant \( C \) shifts the parabola along the x-axis.
• Understanding parabolas helps in many applications like projectile motion.

When sketching, adjusting \( C \) shows various parabolas intersecting the exponential curves at right angles. This can clarify how families of curves relate in geometry.