Advanced High-School Mathematics

SECTION 5.5 Differential Equations 313

y

x+ 1

=

∫ xdx

(x+ 1)^2

=

∫ (x+ 1−1)dx

(x+ 1)^2

=

∫ Ñ 1

x+ 1

−

1

(x+ 1)^2

é

dx

= ln(x+ 1) +

1

x+ 1

+C

It follows, therefore, that

y = (x+ 1) ln(x+ 1) +c(x+ 1),

wherecis an arbitrary constant.

Exercises

1. Solve the following first-order ODE.
   (a) xy′+ 2y= 2x^2 , y(1) = 0
   (b) 2x^2 y′+ 4xy=e−x, y(2) = 1.
   (c) xy′+ (x−2)y= 3x^3 e−x, y(1) = 0
   (d) y′lnx+

y

x

=x, y(1) = 0

(e) y′+ (cotx)y= 3 sinxcosx, y(0) = 1

(f) x(x+ 1)y′−y= 2x^2 (x+ 1), y(2) = 0

2. The first-orderBernoulliODE are of the form

y′+p(x)y=q(x)yn,

wherenis any number other than 1. Show that the substitution

u=y^1 −n brings the above Bernoulli equation into the first-order

linear ODE

1

1 −n

u′+p(x)u=q(x).