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