Differential Equations

To find the particular integral of the differential equation , we'll use the method of undetermined coefficients.

1. Given Differential Equation:
where .

2. Find the Particular Integral (P.I.):
For an exponential forcing function , the P.I. is given by: Here, and .

3. Evaluate :

4. Compute the Particular Integral:
Since , we have:

5. Verification:
Let's verify by substituting back into the differential equation: which matches the right-hand side of the original equation.

6. Comparing with Given Options:
The correct particular integral is:

Final Answer:
The particular integral is .