The particular solution of the given differential equation below is _________

   **y'' + y' = 6x + 12e2x** 

To find the particular solution for the differential equation:

\[
y'' + y' = 6x + 12e^{2x}
\]

We'll follow these steps:

### Step 1: Solve for the **homogeneous equation**
The homogeneous equation associated with this is:

\[
y'' + y' = 0
\

The particular solution of the given differential equation below is y'' + y' = 6x + 12e^2x

Find the particular solution to the differential equation y'' + y' = 6x + 12e^{2x}. This problem explores key concepts in solving differential equations, such as the method of undetermined coefficients and homogeneous equations. Suitable for undergraduate calculus students.

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