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7.4. SEPARABLE DIFFERENTIAL EQUATIONS
where k is a constant of proportionality.
(a) Suppose you measure that the coffee is cooling at one degree per minute at
the time the coffee is brought into the room. Use the differential equation to
determine the value of the constant k.
(b) Find all the solutions of this differential equation.
(c) What happens to all the solutions as t → ∞? Explain how this agrees with your
intuition.
(d) What is the temperature of the cup of coffee after 20 minutes?
(e) How long does it take for the coffee to cool to 80 ◦ ?
⊳
Activity 7.10.
Solve each of the following differential equations or initial value problems.
(a)
dy
dt
− (2 − t)y = 2 − t
(b)
1
t
dy
dt
= e
t 2 −2y
(c) y ′ = 2y + 2, y(0) = 2
(d) y ′ = 2y 2 , y(−1) = 2
(e)
dy
dt
=
−2t y
t 2 + 1
, y(0) = 4
⊳
Summary
In this section, we encountered the following important ideas:
• A separable differential equation is one that may be rewritten with all occurrences of the
dependent variable multiplying the derivative and all occurrences of the independent
variable on the other side of the equation.
• We may find the solutions to certain separable differential equations by separating
variables, integrating with respect to t, and ultimately solving the resulting algebraic
equation for y.
• This technique allows us to solve many important differential equations that arise in
the world around us. For instance, questions of growth and decay and Newton’s Law of
Cooling give rise to separable differential equations. Later, we will learn in Section 7.6
that the important logistic differential equation is also separable.
7.4. SEPARABLE DIFFERENTIAL EQUATIONS
where k is a constant of proportionality.
(a) Suppose you measure that the coffee is cooling at one degree per minute at
the time the coffee is brought into the room. Use the differential equation to
determine the value of the constant k.
(b) Find all the solutions of this differential equation.
(c) What happens to all the solutions as t → ∞? Explain how this agrees with your
intuition.
(d) What is the temperature of the cup of coffee after 20 minutes?
(e) How long does it take for the coffee to cool to 80 ◦ ?
⊳
Activity 7.10.
Solve each of the following differential equations or initial value problems.
(a)
dy
dt
− (2 − t)y = 2 − t
(b)
1
t
dy
dt
= e
t 2 −2y
(c) y ′ = 2y + 2, y(0) = 2
(d) y ′ = 2y 2 , y(−1) = 2
(e)
dy
dt
=
−2t y
t 2 + 1
, y(0) = 4
⊳
Summary
In this section, we encountered the following important ideas:
• A separable differential equation is one that may be rewritten with all occurrences of the
dependent variable multiplying the derivative and all occurrences of the independent
variable on the other side of the equation.
• We may find the solutions to certain separable differential equations by separating
variables, integrating with respect to t, and ultimately solving the resulting algebraic
equation for y.
• This technique allows us to solve many important differential equations that arise in
the world around us. For instance, questions of growth and decay and Newton’s Law of
Cooling give rise to separable differential equations. Later, we will learn in Section 7.6
that the important logistic differential equation is also separable.