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In this section, we will learn about certain differential equations that can be solved explicitly.
1.
9
DIFFERENTIAL EQUATIONS
2.
DIFFERENTIAL EQUATIONS
We have looked at first-order differential
equations from a geometric point of view
(direction fields) and from a numerical point
of view (Euler’s method).
What about the symbolic point of view?
3.
DIFFERENTIAL EQUATIONS
It would be nice to have an explicit
formula for a solution of a differential
Unfortunately, that is not always possible.
4.
DIFFERENTIAL EQUATIONS
9.3
Separable Equations
In this section, we will learn about:
Certain differential equations
that can be solved explicitly.
5.
SEPARABLE EQUATION
A separable equation is a first-order
differential equation in which the expression
for dy/dx can be factored as a function of
x times a function of y.
In other words, it can be written
in the form dy
= g ( x) f ( y )
dx
6.
SEPARABLE EQUATIONS
The name separable comes from
the fact that the expression on the right side
can be “separated” into a function of x
and a function of y.
7.
SEPARABLE EQUATIONS Equation 1
Equivalently, if f(y) ≠ 0, we could write
dy g ( x)
=
dx h( y )
where h( y ) = 1/ f ( y )
8.
SEPARABLE EQUATIONS
To solve this equation, we rewrite it in
the differential form
h(y) dy = g(x) dx
so that:
All y’s are on one side of the equation.
All x’s are on the other side.
9.
SEPARABLE EQUATIONS Equation 2
Then, we integrate both sides
of the equation:
∫h ( y ) dy = ∫g ( x ) dx
10.
SEPARABLE EQUATIONS
Equation 2 defines y implicitly as
a function of x.
In some cases, we may be able to solve for y
in terms of x.
11.
SEPARABLE EQUATIONS
We use the Chain Rule to justify this
If h and g satisfy Equation 2,
then
d d
dx
( )
∫ h( y) dy = dx (∫ g ( x) dx )
12.
SEPARABLE EQUATIONS
Thus,
d dy
dy
( )
∫ h( y) dy dx = g ( x)
This gives:
dy
h( y ) = g ( x )
dx
Thus, Equation 1 is satisfied.
13.
SEPARABLE EQUATIONS Example 1
a. Solve the differential equation
2
dy x
= 2
dx y
b. Find the solution of this equation that
satisfies the initial condition y(0) = 2.
14.
SEPARABLE EQUATIONS Example 1 a
We write the equation in terms of differentials
and integrate both sides:
y2 dy = x2 dx
∫ y2 dy = ∫ x2 dx
⅓y3 = ⅓x3 + C
where C is an arbitrary constant.
15.
SEPARABLE EQUATIONS Example 1 a
We could have used a constant C1 on
the left side and another constant C2 on
the right side.
However, then, we could combine
these constants by writing C = C2 – C1.
16.
SEPARABLE EQUATIONS Example 1 a
Solving for y, we get:
3 3
y = x + 3C
We could leave the solution like this or we could
write it in the form
3 3
y= x +K
where K = 3C.
Since C is an arbitrary constant, so is K.
17.
SEPARABLE EQUATIONS Example 1 b
If we put x = 0 in the general solution in (a),
we get: 3
y (0) = K
To satisfy the initial condition y(0) = 2,
3
we must have K = 2 , and so K = 8.
So, the solution of the initial-value problem
is:
3 3
y = x +8
18.
SEPARABLE EQUATIONS
The figure shows graphs of several members
of the family of solutions of the differential
equation in Example 1.
The solution of
the initial-value problem
in (b) is shown in red.
19.
SEPARABLE EQUATIONS Example 2
Solve the differential equation
2
dy 6x
=
dx 2 y + cos y
20.
SEPARABLE EQUATIONS E. g. 2—Equation 3
Writing the equation in differential form
and integrating both sides, we have:
(2y + cos y) dy = 6x2 dx
∫ (2y + cos y) dy = ∫ 6x2 dx
y2 + sin y = 2x3 + C
where C is a constant.
21.
SEPARABLE EQUATIONS Example 2
Equation 3 gives the general solution
In this case, it’s impossible to solve the equation
to express y explicitly as a function of x.
22.
SEPARABLE EQUATIONS
The figure shows the graphs of several
members of the family of solutions of
the differential equation in Example 2.
As we look at
the curves from left
to right, the values
of C are:
3, 2, 1, 0, -1, -2, -3
23.
SEPARABLE EQUATIONS Example 3
Solve the equation
y’ = x2y
First, we rewrite the equation
using Leibniz notation:
dy 2
=x y
dx
24.
SEPARABLE EQUATIONS Example 3
If y ≠ 0, we can rewrite it in differential
notation and integrate:
dy
= x 2 dx y ≠0
y
dy 2
∫ y = ∫x dx
x3
ln y = + C
3
25.
SEPARABLE EQUATIONS Example 3
The equation defines y implicitly as a function
of x.
However, in this case, we can solve explicitly
for y.
y =e ln y
=e
( )
x3 / 3 + C
C x
=e e
3
/3
C x3 / 3
Hence, y = ±e e
26.
SEPARABLE EQUATIONS Example 3
We can easily verify that the function y = 0
is also a solution of the given differential
So, we can write the general solution in the form
x3 / 3
y = Ae
where A is an arbitrary constant (A = eC,
or A = –eC, or A = 0).
27.
SEPARABLE EQUATIONS
The figure shows a direction field for
the differential equation in Example 3.
Compare it with the next
figure, in which we use
x3 / 3
the equation y = Ae
to graph solutions for
several values of A.
28.
SEPARABLE EQUATIONS
If you use the direction field to sketch solution
curves with y-intercepts 5, 2, 1, –1, and –2,
they will resemble the curves in the figure.
29.
SEPARABLE EQUATIONS Example 4
In Section 9.2, we modeled the current I(t)
in this electric circuit by the differential
dI
L + RI = E (t )
dt
30.
SEPARABLE EQUATIONS Example 4
Find an expression for the current in a circuit
The resistance is 12 Ω.
The inductance is 4 H.
A battery gives a constant voltage of 60 V.
The switch is turned on when t = 0.
What is the limiting
value of the current?
31.
SEPARABLE EQUATIONS Example 4
With L = 4, R = 12 and E(t) = 60,
The equation becomes:
dI dI
4 + 12 I = 60 or = 15 −3I
dt dt
The initial-value problem is:
dI
= 15 − 3I I (0 ) = 0
dt
32.
SEPARABLE EQUATIONS Example 4
We recognize this as being separable.
We solve it as follows:
dI
∫ 15 − 3I = ∫ dt (15 − 3I ≠ 0)
− 13 ln 15 − 3I = t + C
−3(t +C )
15 − 3I = e
−3C −3t −3t
15 − 3I = ± e e = Ae
−3t
I = 5 − Ae1
3
33.
SEPARABLE EQUATIONS Example 4
Since I(0) = 0, we have:
5 – ⅓A = 0
So, A = 15 and the solution is:
I(t) = 5 – 5e-3t
34.
SEPARABLE EQUATIONS Example 4
The limiting current, in amperes, is:
lim I (t ) = lim (5 − 5e −3t
)
t →∞ t →∞
−3t
= 5 − 5lim e
t →∞
= 5−0
=5
35.
SEPARABLE EQUATIONS
The figure shows how the solution
in Example 4 (the current) approaches
its limiting value.
36.
SEPARABLE EQUATIONS
Comparison with the other figure (from
Section 9.2) shows that we were able to
draw a fairly accurate solution curve from
the direction field.
37.
ORTHOGONAL TRAJECTORY
An orthogonal trajectory of a family of curves
is a curve that intersects each curve of the
family orthogonally—that is, at right angles.
38.
ORTHOGONAL TRAJECTORIES
Each member of the family y = mx of straight
lines through the origin is an orthogonal
trajectory of the family x2 + y2 = r2 of concentric
circles with
center the origin.
We say that the two
families are orthogonal
trajectories of each other.
39.
ORTHOGONAL TRAJECTORIES Example 5
Find the orthogonal trajectories of
the family of curves x = ky2, where k
is an arbitrary constant.
40.
ORTHOGONAL TRAJECTORIES Example 5
The curves x = ky2 form a family
of parabolas whose axis of symmetry
is the x-axis.
The first step is to find a single differential equation
that is satisfied by all members of the family.
41.
ORTHOGONAL TRAJECTORIES Example 5
If we differentiate x = ky2, we get:
dy dy 1
1 = 2ky or =
dx dx 2ky
This differential equation depends on k.
However, we need an equation that is valid
for all values of k simultaneously.
42.
ORTHOGONAL TRAJECTORIES Example 5
To eliminate k, we note that:
From the equation of the given
general parabola x = ky2, we have k = x/y2.
43.
ORTHOGONAL TRAJECTORIES Example 5
Hence, the differential equation can be
written as: dy 1 1
= =
dx 2ky 2 x y
y2
dy y
or =
dx 2 x
This means that the slope of the tangent line at
any point (x, y) on one of the parabolas is: y’ = y/(2x)
44.
ORTHOGONAL TRAJECTORIES
On an orthogonal trajectory, the slope
of the tangent line must be the negative
reciprocal of this slope.
So, the orthogonal trajectories must satisfy
the differential equation
dy 2x
=−
dx y
45.
ORTHOGONAL TRAJECTORIES E. g. 5—Equation 4
The differential equation is separable.
We solve it as follows:
∫y dy = −∫2 x dx
y2
= −x 2 + C
2
2
2 y
x + =C
2
where C is an arbitrary positive constant.
46.
ORTHOGONAL TRAJECTORIES Example 5
Thus, the orthogonal trajectories are
the family of ellipses given by Equation 4
and sketched here.
47.
ORTHOGONAL TRAJECTORIES IN PHYSICS
Orthogonal trajectories occur in various
branches of physics.
In an electrostatic field, the lines of force are
orthogonal to the lines of constant potential.
The streamlines in aerodynamics are orthogonal
trajectories of the velocity-equipotential curves.
48.
MIXING PROBLEMS
A typical mixing problem involves a tank
of fixed capacity filled with a thoroughly mixed
solution of some substance, such as salt.
A solution of a given concentration enters the tank
at a fixed rate.
The mixture, thoroughly stirred, leaves at a fixed rate,
which may differ from the entering rate.
49.
MIXING PROBLEMS
If y(t) denotes the amount of substance in
the tank at time t, then y’(t) is the rate at which
the substance is being added minus the rate
at which it is being removed.
The mathematical description of this situation often
leads to a first-order separable differential equation.
50.
MIXING PROBLEMS
We can use the same type of reasoning
to model a variety of phenomena:
Chemical reactions
Discharge of pollutants into a lake
Injection of a drug into the bloodstream
51.
MIXING PROBLEMS Example 6
A tank contains 20 kg of salt dissolved
in 5000 L of water.
Brine that contains 0.03 kg of salt per liter of water
enters the tank at a rate of 25 L/min.
The solution is kept thoroughly mixed and drains
from the tank at the same rate.
How much salt remains in the tank after half an hour?
52.
MIXING PROBLEMS Example 6
Let y(t) be the amount of salt (in kilograms)
after t minutes.
We are given that y(0) = 20 and we want to
find y(30).
We do this by finding a differential equation
satisfied by y(t).
53.
MIXING PROBLEMS Equation 5
Note that dy/dt is the rate of change of
the amount of salt.
Thus, dy
= (rate in ) − (rate out )
dt
‘Rate in’ is the rate at which salt enters the tank.
‘Rate out’ is the rate at which it leaves the tank.
54.
RATE IN Example 6
We have:
⎛ kg ⎞⎛ L ⎞
rate in = ⎜0.03 ⎟⎜25 ⎟
⎝ L ⎠⎝ min ⎠
kg
= 0.75
min
55.
MIXING PROBLEMS Example 6
The tank always contains 5000 L
of liquid.
So, the concentration at time t is y(t)/5000
(measured in kg/L).
56.
RATE OUT Example 6
As the brine flows out at a rate of 25 L/min,
we have:
⎛ y (t ) kg ⎞⎛ L ⎞
rate out = ⎜ ⎟⎜25 ⎟
⎝ 5000 L ⎠⎝ min ⎠
y (t ) kg
=
200 min
57.
MIXING PROBLEMS Example 6
Thus, from Equation 5, we get:
dy y (t ) 150 −y (t )
= 0.75 − =
dt 200 200
Solving this separable differential equation,
we obtain:
dy dt
∫150 −y = ∫200
t
−ln 150 −y = +C
200
58.
MIXING PROBLEMS
Since y(0) = 20, we have:
–ln 130 = C
t
−ln 150 −y = −ln130
200
59.
MIXING PROBLEMS Example 6
−t / 200
150 −y = 130e
y(t) is continuous and y(0) = 20, and the right side
is never 0.
We deduce that 150 – y(t) is always positive.
60.
MIXING PROBLEMS Example 6
Thus, |150 – y| = 150 – y.
−t / 200
y (t ) = 150 −130e
The amount of salt after 30 min is:
−30 200
y (30) = 150 −130e ≈38.1 kg
61.
MIXING PROBLEMS Example 6
Here’s the graph of the function y(t)
of Example 6.
Notice that, as time
goes by, the amount
of salt approaches
150 kg.
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