# 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)