Nonhomogeneous linear equations with constant coefficients

2-Nonhomogeneous linear equations with constant coefficients
A second order nonhomogeneous equation with constant coefficients is written as


I- Undetermined Coefficients
We use the method of undetermined coefficients for finding particular solutions when is one of the forms shown in table.
Form of
Notes





or


We write the general solution of as the sum of the homogeneous and particular solutions
Example 1:
Solution: First, we solve the homogeneous equation. The characteristic equation is



Upon substitution into the differential equation, we have




Example 2:
Solution: First, we solve the homogeneous equation. The characteristic equation is




Upon substitution into the differential equation, we have


Equating coefficient on the left side with the coefficient in the right side we get

By the same way equate coefficient :
And


Example 3:
Solution: First, we solve the homogeneous equation. The characteristic equation is


Second, we find a particular solution of the nonhomogeneous equation. The form of the particular solution is chosen such that the exponential will cancel out of both sides of the differential equation. We choose




Example 4:
Solution:








Example 5:
Solution: The characteristic equation is



Upon substitution into the differential equation, we obtain







Example 6:
Solution: The characteristic equation is









Example 7:
Solution:










Example 8:
Solution:






Upon substitution into the differential equation, we obtain






For each of the following problem solve the differential equation