7.4. SEPARABLE DIFFERENTIAL EQUATIONS
421
Solution. We begin by separating the variables and writing
y
2 dy
dt
= t.
Integrating both sides of the equation with respect to the independent variable t shows
that
y
2 dy
dt
dt =
t dt.
Next, we notice that the left-hand side allows us to change the variable of antidifferentiation 4 from t to y. In particular, dy =
dy
dt dt, so we now have
y
2 dy =
t dt.
This most recent equation says that two families of antiderivatives are equal to one another.
Therefore, when we find representative antiderivatives of both sides, we know they must
differ by arbitrary constant C. Antidifferentiating and including the integration constant
C on the right, we find that
y 3
3
=
t 2
2
+ C.
Again, note that it is not necessary to include an arbitrary constant on both sides of the
equation; we know that y 3 /3 and t 2 /2 are in the same family of antiderivatives and must
therefore differ by a single constant.
Finally, we may now solve the last equation above for y as a function of t, which gives
y(t) =
3
3
2
t 2 + 3C.
Of course, the term 3C on the right-hand side represents 3 times an unknown constant. It
is, therefore, still an unknown constant, which we will rewrite as C. We thus conclude that
the funtion
y(t) =
3
3
2
t 2 + C
is a solution to the original differential equation for any value of C.
Notice that because this solution depends on the arbitrary constant C, we have found
an infinite family of solutions. This makes sense because we expect to find a unique
solution that corresponds to any given initial value.
For example, if we want to solve the initial value problem
dy
dt
=
t
y 2 , y(0) = 2,
4 This is why we required that the left-hand side be written as a product in which dy/dt is one of the terms.
421
Solution. We begin by separating the variables and writing
y
2 dy
dt
= t.
Integrating both sides of the equation with respect to the independent variable t shows
that
y
2 dy
dt
dt =
t dt.
Next, we notice that the left-hand side allows us to change the variable of antidifferentiation 4 from t to y. In particular, dy =
dy
dt dt, so we now have
y
2 dy =
t dt.
This most recent equation says that two families of antiderivatives are equal to one another.
Therefore, when we find representative antiderivatives of both sides, we know they must
differ by arbitrary constant C. Antidifferentiating and including the integration constant
C on the right, we find that
y 3
3
=
t 2
2
+ C.
Again, note that it is not necessary to include an arbitrary constant on both sides of the
equation; we know that y 3 /3 and t 2 /2 are in the same family of antiderivatives and must
therefore differ by a single constant.
Finally, we may now solve the last equation above for y as a function of t, which gives
y(t) =
3
3
2
t 2 + 3C.
Of course, the term 3C on the right-hand side represents 3 times an unknown constant. It
is, therefore, still an unknown constant, which we will rewrite as C. We thus conclude that
the funtion
y(t) =
3
3
2
t 2 + C
is a solution to the original differential equation for any value of C.
Notice that because this solution depends on the arbitrary constant C, we have found
an infinite family of solutions. This makes sense because we expect to find a unique
solution that corresponds to any given initial value.
For example, if we want to solve the initial value problem
dy
dt
=
t
y 2 , y(0) = 2,
4 This is why we required that the left-hand side be written as a product in which dy/dt is one of the terms.