7.4. SEPARABLE DIFFERENTIAL EQUATIONS
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7.4 Separable differential equations
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a separable differential equation?
• How can we find solutions to a separable differential equation?
• Are some of the differential equations that arise in applications separable?
Introduction
In Sections 7.2 and 7.3, we have seen several ways to approximate the solution to an
initial value problem. Given the frequency with which differential equations arise in the
world around us, we would like to have some techniques for finding explicit algebraic
solutions of certain initial value problems. In this section, we focus on a particular class of
differential equations (called separable) and develop a method for finding algebraic formulas
for solutions to these equations.
A separable differential equation is a differential equation whose algebraic structure
permits the variables present to be separated in a particular way. For instance, consider
the equation
dy
dt
= t y.
We would like to separate the variables t and y so that all occurrences of t appear on the
right-hand side, and all occurrences of y appears on the left and multiply dy/dt. We may
do this in the preceding differential equation by dividing both sides by y:
1
y
dy
dt
= t.
Note particularly that when we attempt to separate the variables in a differential equation,
we require that the left-hand side be a product in which the derivative dy/dt is one term.
Not every differential equation is separable. For example, if we consider the equation
dy
dt
= t − y,
it may seem natural to separate it by writing
y +
dy
dt
= t.