# Particular Solution of a Nonhomogeneous Linear Second-Order Differential Equation with Constant Coefficients

Particular Solution of a Nonhomogeneous Linear Second-Order Differential Equation with Constant Coefficients

This Demonstration shows the method of undetermined coefficients for a nonhomogeneous differential equation of the form with , , , and constants. If , then the form of the particular solution is . If and , the particular solution is of the form . If and , the particular solution is of the form (cx+d).

y''+py'+q=ax+b

p

q

a

b

q≠0

cx+d

q=0

p≠0

x(cx+d)

q=0

p=0

2

x

The second part shows the solution of a linear nonhomogeneous second-order differential equation of the form . Let be a root of the corresponding characteristic equation. If , the particular solution is of the form (ccos(bx)+dsin(bx)). If and , the form is . If has multiplicity 2, then is a real number and the form of particular solution is .

y''+py'+qy=cos(bx)

ax

e

r

r≠a±bi

ax

e

r=a±bi

b≠0

x(ccos(bx)+dsin(bx))

ax

e

r

r=a

c

2

x

ax

e