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APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS
DR. HUDA ALRASHDI
1. OnrHoconaL Trae
oRtEs
Suppose that we have a family of two curves given by
fly =0, and g(x,y.k) =
“These two curves are said to be orthogonal at a point if and only is their tangent
lines T; and T; are perpendicular at the point of intersection,
Definition 1. orthogonal trajectories are two families of curves that always intersect
penpendicularly
In other words, an orthogonal trajectory is any one curve that intersects every
curve of another family at right angles.
‘Now we will show a method to how to find the fanily of orthogonal trajectories
2. METHOD,
D
© STEP 1 Find the differential equation for the given family of curves, by
differentiating f(x, y.c)
Given a family of curve Fx, y,¢)
© STEP 2 Find the differential equation of the Orthogenal Trajectories
dy 1 dr
Fen ene
dn
° STEPS
parating, variables and integrating the above differential equation,
we get the algebraie equation of the family of orthogonal trajectories,
Example 1. Find the orthogonal trajectories of all parabolas with vertices at the
origise ared foct on the r-arts
1)
Solution We con rewrite the equation (1) as follows
a ‘9
eT da, (2) a DR. HUDA ALRASHDI
© Differentiate equation (2)
0
u
2r
© Differential equation of the orthogonal trajectories
dy dy Be
ee
dr ftx,y) ar y
© Solving the differential equation by metho of separation of variables
dye
ay
~yly = Bede
Deeb + yy = 0
[ries fvto= fo
L
+ rae C, Cis an integration constant,
Dr? fey? where Cy = 20.
Hence the orthogonal trajectories are
art +yr=b, family of ellipse
Example 2. Find the orthagonal trajectories of
wt te)+2= 8)
Salution
# Differentiate equation. (3):
ty ea =
ha? +e) + (2x) =0
4
dr +e ©)
Ta find c: From equation (3) me have c= =2=##, Substitute an (4) we get:
dy APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS a
dy =2ry?
de Py + (-2— 2p)
ay ott
ar =2
ae
a 7H
© Differential equation of the orthogonal trajectories
dy 1 dy 1
de Teg} de
© Solving the differential eqaution by method of separation of variables
dy 1
de
widy
[r= [jae
2
In|x|+C, Cas an integration constant,
yf = Bin |x| +30,
Hence the orthogonal trajectories are
yh = Bln |x| 4+Cy, where Cy = 3C
Example 3. Find the orthogonal drajectories of
(5)
Solution Pind c:
# Differentiate equation (5). DR. HUDA ALRASHDI
dy
VE oe
dy _ yp t+a
dz Bry?
© Differential equation of the orthogonal trajectories
de dr tty?
Galen =F =F
de lyn
aq IGT e
We know thal © = 4(z + ») is homogeneous equation because f(x,y) =
aly
+ £) 18 homogencous junction,
a(t oe
sttayta) = —3( + 2) = 8709)
© Solving the differential equation by method of homogeneous equation,
Let
So,
—2yvde = (vu + Daly APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS
In 30? + 1
In|3u? +1
[au" + 1
Bu? +1
Be? 41 = Ky where K = be
35 +1=k4
v y
K
ety ty
Hence the orthogonal trajectories are
(dx? + yt) = K
3. EXERCISES
Find the orthogonal trajectories of the following equations:
(1) wer? + I) = cr,
(2) yr? + 9")
(3) w= csina,
rl — cr).
