Lecture Notes in Differential Equations - Bruce E. Shapiro

19
Since it is not possible to solve this equation for y as a function of t, is is
common to rearrange it as a function equal to a constant:
e y − y 2 − 1 2 t2 = C (3.17)
Sometimes this inability to find an explicit formula for y(t) means that the
relationship between y and t is not a function, but is instead multi-valued,
as in the following example where we can use our knowledge of analytic
geometry to learn more about the solutions.
Example 3.3. Find the general solution of
dy
dt = − 4t
9y
Rearranging and integrating:
∫ ∫
9ydy = −
Dividing both sides by 36,
Dividing by C
(3.18)
4tdt (3.19)
9
2 y2 = −2t 2 + C (3.20)
9y 2 + 4t 2 = C (Different C) (3.21)
y 2
4 + t2 9
= C (Different C) (3.22)
y 2
4C + t2
9C = 1 (3.23)
which is the general form of an ellipse with axis 2 √ C parallel to the y
axis and axis 3 √ C parallel to the y axis. Thus the solutions are all ellipses
around the origin, and these cannot be solved explicitly as a function
because the formula is multi-valued.
Example 3.4. Solve 2 (1 + x)dy − ydx − 0. Ans: y = c(1 + x)
Example 3.5. Solve 3
Ans:
2 Zill Example 2.2.1
3 Zill Example 2.2.3
dy
dx = y2 − 4 (3.24)
y = 2 1 + Ce4x
or y = ±2 (3.25)
1 − Ce4x