Math 225 Differential Equations Notes Chapter 2

2.3 Linear Equations
Recall from Section 1.1 that a linear first-order equation is of the
form:
a 1 (x) dy
dx + a 0(x)y = b(x)
There are two situation for which the solution is quite immediate.
If a 0 (x) := 0 then
a 1 (x) dy
dx = b(x)
∫ b(x)
y(x) =
a 1 (x) dx + C
The second situation is if a 0 (x) happens to be equal to the derivative
of a 1 (x) In this case:
a 1 (x)y ′ + a 0 (x)y = a 1 (x)y ′ + a ′ 1(x)y = d
dx [a 1(x)y]
d
dx [a 1(x)y] = b(x)
∫
a 1 (x)y = b(x)dx + C
y(x) = 1 [∫
a 1 (x)
]
b(x)dx + C
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