Question

Differential equations will have solution of the form

Answer 1

The most general linear second order differential equation is in the form.

p
(
t
)
y
′′
+
q
(
t
)
y
′
+
r
(
t
)
y
=
g
(
t
)

(1)

In fact, we will rarely look at non-constant coefficient linear second order differential equations. In the case where we assume constant coefficients we will use the following differential equation.

a
y
′′
+
b
y
′
+
c
y
=
g
(
t
)

(2)

Where possible we will use
(1)
just to make the point that certain facts, theorems, properties, and/or techniques can be used with the non-constant form. However, most of the time we will be using
(2)
as it can be fairly difficult to solve second order non-constant coefficient differential equations.

Initially we will make our life easier by looking at differential equations with
g
(
t
)
=
0
. When
g
(
t
)
=
0
we call the differential equation homogeneous and when
g
(
t
)
≠
0
we call the differential equation nonhomogeneous.

So, let’s start thinking about how to go about solving a constant coefficient, homogeneous, linear, second order differential equation. Here is the general constant coefficient, homogeneous, linear, second order differential equation.

a
y
′′
+
b
y
′
+
c
y
=
0
It’s probably best to start off with an example. This example will lead us to a very important fact that we will use in every problem from this point on. The example will also give us clues into how to go about solving these in general.
I hope this will help you