Please answer the attached question in pic 1. Pic 2 is the solution - please explain the steps. Thank you. 42. The hyperbolas with F(x, y) = y²/x = C are thesolid curves in the following figure. The orthogonalfamily must satisfydy2xdxдуaxyThe solution to this separable equation is found to begiven implicitly by G(x, y) = 2x² + y² = C. Thesecurves are the dashed ellipses in the accompanyingfigure. They do appear to be orthogonal.1> 0-1-2-2-1 0 1 2 C defines a family ofAn equation of the form F(x, y)curves in the plane. Furthermore, we know these curves arethe integral curves of the differential equationdyƏFdy = 0дуƏFdx +axdFordxaxdy(6.44)A family of curves is said to be orthogonal to a second fam-ily if each member of one family intersects all members of theother family at right angles. For example, the families y = mxand x? + y? = c² are orthogonal. For a curve y = y(x) to beeverywhere orthogonal to the curves defined by F(x, y) = Cits derivative must be the negative reciprocal of that in (6.44),ordyƏFƏFdxдуaxThe family of solutions to this differential equation are orthog-onal to the family defined by F(x, y) = C.42. Find the family of curves that is orthogonal to the fam-ily defined by the equation y?depicting the orthogonality of the two families.= cx and provide a sketch

42. given curve is derivative of the curve is…

Answered: 42. Find the family of curves that is…

42. Find the family of curves that is orthogonal to the fam- ily defined by the equation y? = cx and provide a sketch depicting the orthogonality of the two families.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.

Question

Please answer the attached question in pic 1. Pic 2 is the solution - please explain the steps. Thank you.

42. The hyperbolas with F(x, y) = y²/x = C are the
solid curves in the following figure. The orthogonal
family must satisfy
dy
2x
dx
ду
ax
y
The solution to this separable equation is found to be
given implicitly by G(x, y) = 2x² + y² = C. These
curves are the dashed ellipses in the accompanying
figure. They do appear to be orthogonal.
1
> 0
-1
-2
-2
-1 0 1 2

C defines a family of
An equation of the form F(x, y)
curves in the plane. Furthermore, we know these curves are
the integral curves of the differential equation
dy
ƏF
dy = 0
ду
ƏF
dx +
ax
dF
or
dx
ax
dy
(6.44)
A family of curves is said to be orthogonal to a second fam-
ily if each member of one family intersects all members of the
other family at right angles. For example, the families y = mx
and x? + y? = c² are orthogonal. For a curve y = y(x) to be
everywhere orthogonal to the curves defined by F(x, y) = C
its derivative must be the negative reciprocal of that in (6.44),
or
dy
ƏF
ƏF
dx
ду
ax
The family of solutions to this differential equation are orthog-
onal to the family defined by F(x, y) = C.
42. Find the family of curves that is orthogonal to the fam-
ily defined by the equation y?
depicting the orthogonality of the two families.
= cx and provide a sketch

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