Week 8 Lecture Notes, Lec01
9.5. Linear Equations
A first-order linear differential equation is one that can be put into the form:
??
??
+ ?(?)? = ?(?)
Where
?(?)
and
?(?)
are continuous functions on a given interval. We call this the
standard form
of a linear differential equation and show it with (*). We can also
write it as:
?′ + ?(?)? = ?(?)
Example.
?
2
?′ + 2?? = 4
Solution.
The RHS seems to be the derivative of
?
2
?
. So:
(?
2
?)′ = 4 → ?
2
? = 4? + 𝐶 → ? =
4? + 𝐶
?
2
If the equation was introduced in the standard form:
?′ +
2
?
? =
4
?
2
Then we had to first, multiply
?
2
by both sides so that we could write the RHS as
the derivative of
?
2
?
.