Answer:
Step-by-step explanation:
The general solution will be the sum of the complementary solution and the particular solution:
In order to find the complementary solution you need to solve:
Using the characteristic equation, we may have three cases:
Real roots:
Repeated roots:
Complex roots:
Hence:
Solving for :
Since we got complex roots, the complementary solution will be given by:
Now using undetermined coefficients, the particular solution is of the form:
Note: was multiplied by x to account for
and
in the complementary solution.
Find the second derivative of in order to find the constants
and
:
Substitute the particular solution into the differential equation:
Simplifying:
Equate the coefficients of and
on both sides of the equation:
So:
Substitute the value of the constants into the particular equation:
Therefore, the general solution is: