Lecture Notes in Differential Equations - Bruce E. Shapiro

20 LESSON 3. SEPARABLE EQUATIONS
Example 3.6. Solve 4
Ans:
(e 2y − y) cos x dy
dx = ey sin 2x, y(0) = 0 (3.26)
e y + ye −y + e −y = 4 − 2 cos x (3.27)
Example 3.7. Solve 5
Ans:
dy
dx = e−x2 , y(3) = 5 (3.28)
y(x) = 5 +
∫ x
3
e −t2 dt (3.29)
Definition 3.1. The Error Function erf(x) is defined as
A plot of erf(x) is given in figure 4.2.
erf(x) = √ 2 ∫ x
e −t2 dt (3.30)
π
Example 3.8. Rewrite the solution to example 3.7 in terms of erf(x)
0
√ π
y(x) = 5 + (erf(x) − erf(3)) (3.31)
2
Sometimes it is not so easy to tell by looking at an equation if it is separable
because it may need to be factored before the variables can be separated.
There is a test that we can use that will sometimes help us to disentangle
these variables. To derive this test, we will rearrange the general separable
equation as follows
4 Zill Example 2.2.4
5 Zill Example 2.2.5
dy
= a(t)b(y)
dt
(3.32)
dy
= a(t)dt
b(y)
(3.33)
dy
− a(t)dt = 0
b(y)
(3.34)
N(y)dy + M(t)dt = 0 (3.35)