Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 3
Separable Equations
An ODE is said to be separable if the parts that depend on t and y can
be separated to the different sides of the equation. This makes it possible
to integrate each side separable. 1 Specifically, an equation is separable if it
can be written is
dy
= a(t)b(y) (3.1)
dt
for some function a(t) that depends only on t, but not on y, and some
function b(y) that depends only on y and and not on t. If we multiply
through by dt and divide by b(y) the equation becomes
so we may integrate: ∫ dy
b(y) = ∫
dy
= a(t)dt (3.2)
b(y)
a(t)dt (3.3)
We have already seen many separable equations. Another is given in the
following example.
Example 3.1.
dy
(1
dt = − t )
2
y(0) = 1
y 2
⎫
⎪⎬
⎪⎭
(3.4)
1 In this section of the text, Boyce and DiPrima have chosen to use x rather than t as
the independent variable, probably because it will look more like exact two-dimensional
derivatives of the type you should have seen in Math 250.
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