Solution of Some Second-Order Differential Equations with Constant Coefficients

f(x)	psin(tx)+qcos(tx)

coefficients of DE

a

1

b

0

c

1

p

1

q

1

r

1

t

1

arbitrary constants of integration

k

1

2

k

2

0

2

b

-4ac = -4

The general solution of a second-order linear differential equation with constant coefficients

ay''+by'+cy=f(x)

can be written as the sum of the complementary function

y

H

(red) and a particular solution

y

p

(blue), which depends on

a

,

b

,

c

, and

f(x)

. Particular solutions (magenta) of a differential equation are obtained by varying the arbitrary constants

k

1

and

k

2

, 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.

The sign of the discriminant

Δ=

2

b

-4ac

(whose value is shown beneath the complementary function) determines the form of the complementary function:

if

Δ>0

,

k

1

αx

e

+

k

2

βx

e

,

if

Δ=0

,

αx

e

(

k

1

+

k

2

x)

,

if

Δ<0

,

px

e

(

k

1

cosqx+

k

2

sinqx)

.

You can vary the controls to get special forms of

f(x)

that occur most frequently in practice: zero, trigonometric, polynomial, and exponential functions.