Solution of Some Second-Order Differential Equations with Constant Coefficients
Solution of Some Second-Order Differential Equations with Constant Coefficients
The general solution of a second-order linear differential equation with constant coefficients can be written as the sum of the complementary function (red) and a particular solution (blue), which depends on , , , and . Particular solutions (magenta) of a differential equation are obtained by varying the arbitrary constants and , which also specify a particular complementary function. The complementary function and particular solution are plotted at the bottom, and the general solution is plotted at the top with the summands dashed.
ay''+by'+cy=f(x)
y
H
y
p
a
b
c
f(x)
k
1
k
2
The sign of the discriminant (whose value is shown beneath the complementary function) determines the form of the complementary function:
Δ=-4ac
2
b
if , +,
Δ>0
k
1
αx
e
k
2
βx
e
if , (+x),
Δ=0
αx
e
k
1
k
2
if , (cosqx+sinqx).
Δ<0
px
e
k
1
k
2
You can vary the controls to get special forms of that occur most frequently in practice: zero, trigonometric, polynomial, and exponential functions.
f(x)