Page 34 - Schaum's Outline of Differential Equations

CHAP. 3] CLASSIFICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 17

Multiplying (1) through by x, we obtain

as a second differential form. Multiplying (_/) through by 1/y, we obtain

as a third differential form. Still other differential forms are derived from (_/) by multiplying that equation through
by any other function of x and y.

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3.6. Write the differential equation (xy + 3)dx + (2x - y + l)dy = 0 in standard forn
This equation is in differential form. We rewrite it as

vhic
d
standar
e
s
th
h
ha
f
i
which has the standard form
or
3.7. Determine if the following differential equations are linear:

(a) The equation is linear; here/>(X) = -sin x and q(x) = e*.
(b) The equation is not linear because of the term sin y.
(c) The equation is linear; here/>(X) = 0 and q(x) = 5.
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(d) The equation is not linear because of the term y .
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(e) The equation is not linear because of the y term.
(/) The equation is not linear because of the y 112 term.
x
x
(g) The equation is linear. Rewrite it as y' + (x- e )y = 0 withp(x) =x- e and q(x) = 0.
(h) The equation is not linear because of the 1/y term.

3.8. Determine whether any of the differential equations in Problem 3.7 are Bernoulli equations.
All of the linear equations are Bernoulli equations with n = 0. In addition, three of the nonlinear
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equations, (e), (/) and ( h), are as well. Rewrite (e) as y' = —xy ; it has form (3.4) wilhp(x) = 0, q(x) = —x, and n = 5.
Rewrite (/) as

It has form (3.4) whhp(x) = q(x) = llx and n = 1/2. Rewrite (h) asy' = —xy l whhp(x) = 0, q(x) = —x, and n = —l.

3.9. Determine if the following differential equations are homogeneous: