James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

600 Chapter 9 Differential Equations
arated” into a function of x and a function of y. Equivalently, if f syd ± 0, we could write
1
dy
dx − tsxd
hsyd
where hsyd − 1yf syd. To solve this equation we rewrite it in the differential form
hsyd dy − tsxd dx
The technique for solving separable
differential equations was first used by
James Bernoulli (in 1690) in solving
a problem about pendulums and by
Leibniz (in a letter to Huygens in 1691).
John Bernoulli explained the general
method in a paper published in 1694.
so that all y’s are on one side of the equation and all x’s are on the other side. Then we
inte grate both sides of the equation:
2
y hsyd dy − y tsxd dx
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.
We use the Chain Rule to justify this procedure: If h and t satisfy (2), then
d
Sy hsyd dyD − d Sy tsxd dxD
dx dx
so
and
d
Sy hsyd dyD dy
dy dx − tsxd
hsyd dy
dx − tsxd
Thus Equation 1 is satisfied.
Example 1
(a) Solve the differential equation dy
dx − x 2
y . 2
(b) Find the solution of this equation that satisfies the initial condition ys0d − 2.
SOLUTION
(a) We write the equation in terms of differentials and integrate both sides:
Figure 1 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
part (b) is shown in red.
_3 3
3
y 2 dy − x 2 dx
y y 2 dy − y x 2 dx
1
3 y 3 − 1 3 x 3 1 C
where C is an arbitrary constant. (We could have used a constant C 1 on the left side and
another constant C 2 on the right side. But then we could combine these constants by
writing C − C 2 2 C 1 .)
Solving for y, we get
y − s 3 x 3 1 3C
We could leave the solution like this or we could write it in the form
_3
y − s 3 x 3 1 K
FIGURE 1
where K − 3C. (Since C is an arbitrary constant, so is K.)
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