   Chapter 6.3, Problem 48E

Chapter
Section
Textbook Problem

Finding Orthogonal Trajectories In Exercises 43-48, find the orthogonal trajectories of the family. Use a graphing utility to graph several member, of each family. y = C e x

To determine

To calculate: The orthogonal trajectory of the family of curve y=Cex and use graphing utility to graph the family of curve.

Explanation

Given:

The provided family of curve is y=Cex.

Calculation:

Let us consider the given equation y=Cex …… (1)

Now, differentiate equation (1) with respect to x and get the differential equation,

dydx=Cex …… (2)

Now put value of Cex in equation (2),

dydx=y

As we know that dydx represents the slope of the given family of curves at (x,y), it follows that the orthogonal family has negative reciprocal slope, that is 1y .

So,

dydx=1y

This gives,

ydy=dx

This differential equation is of variable separable form,

f(y)dy=g(x)dx

Now, integrate to get,

ydy=dxy22=x+c1y2=2x+2c1y2+2x=c

Here,2c1=c

Thus, the family of orthogonal trajectories is y2+2x=c

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