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To find the general solution of the differential equation:

\[
y''' + y'' + y' + y = e^{-t} + 4t,
\]

we need to solve both the homogeneous and the non-homogeneous parts.

### Step 1: Solve the Homogeneous Part

The homogeneous part of the differential equation is:

\[
y''' + y'' + y' + y = 0.
\]

The characteristic equation associated with this differential equation is:

\[
r^3 + r^2 + r + 1 = 0.
\]

We will find the roots of this cubic equation to determine the general solution of the homogeneous part.

### Step 2: Solve for the Characteristic Roots

The characteristic equation \( r^3 + r^2 + r + 1 = 0 \) can be factored by trying possible rational roots or using synthetic division. 

Let's proceed with solving for \( r \).

Find the general solution of y''' + y'' + y' + y = e^{-t} + 4t.

Learn how to find the general solution of the third-order differential equation y''' + y'' + y' + y = e^{-t} + 4t. The solution involves solving both the homogeneous and non-homogeneous parts using characteristic equations and the method of undetermined coefficients. Suitable for students at the undergraduate level studying calculus and differential equations.

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